How can I guide and help a new DM through his first campaign (being an experienced DM myself) without having it look like a grab for power or that I think the new DM is incompetent?
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There's a few things you can do to help, but also ways you can unintentionally sabotage the game.
This is the most important thing. You need to show, repeatedly and in front of the other players, that you respect his ruling even when it differs from how you'd handle it. There is nothing that will stop him learning as much as constantly having his mistakes displayed in front of the rest of the group by someone he probably looks up to. The other players will also become unhappy if you consistently argue with him about rulings. When you aren't the DM, but insist on playing by your interpretation of the rules, you become a rules-lawyer, and you don't want that.
Before the game, offer to help with obscure rules
If you let the DM know that you are willing to offer your interpretation of the rules if he asks for them, it will make things easier for him. He may not know all the splatbooks in as much depth as you, and a player who knows all the rules can be a valuable asset. However, unless he specifically asks for your interpretation, assume it's unwelcome. Again, you don't want to undermine his authority, especially in front of the other players.
The reason you should ask before the game is so that it doesn't interrupt the flow of the game, and so it doesn't seem to be in response to any rulings or actions during the game, as this can lower his confidence. If you don't know the rules all that well, and right after a two-hour session (or even worse, in the middle of it) someone with more system mastery offers to help, it seems like they noticed mistakes. However, if it's beforehand, it seems friendlier.
Accept that he will make mistakes
This one sounds easy, but I can tell you from experience it's one of the hardest parts. You have to accept that the other DM will make mistakes that you wouldn't and that he may not even notice. It's not your job to point them out. He will eventually notice on his own if it's a big mistake, and if you've managed to work out a solution, he should too with enough practice. The first few sessions will probably have a few absolutely horrible moments (I've noticed this from every single new DM I've seen run a session), and you want to avoid getting angry at him. He doesn't know any better, and you probably made those mistakes too, or ones very much like them.
Sit back and have fun
It sounds like you haven't been a player in a long, long time. Take advantage of this opportunity to simply sit back, relax and enjoy someone else's campaign. And who knows, you might notice something new that could help you improve too!
Answer questions and shut the heck up!
There is nothing more distracting than someone second-guessing all your decisions as DM. Make it clear that you are there to answer any questions they may have, but respect their decisions. They will learn which decisions are poor ones on their own pretty quickly, and this kind of knowledge tends to stick.
After the game, you can probably get away with pointing out anything that wasn't immediately an obvious mistake to them. But expect resistance and if they won't accept the advice, just back off.
The internet is a great resource. They have already found it if they are cut out to be a DM. They have searched for advice already. If they have not, there isn't anything you can say that will get them to.
Let the guy DM. Unless he specifically asks you for "DM help," focus on just being a player. From your other question it sounds like you are concerned about being able to cut the DM cord, and he's likely to do fine without your help. You're potentially pre-overstepping by asking this question. Heck, he's probably going to do something you can be learning from instead. If the game is really having a problem such that any player might say something, that's fine, but otherwise leave him be. Just because his way of doing something is different doesn't mean it's wrong (or that the other players may not even prefer it to your style). Be available but unless he asks you "oh please be my GM mentor," mind your business.
Be a Good Player
You know all that stuff DMs hate that players do? Don't do that.
Also, a group-aware player that's been in the DM's seat is a huge asset in a campaign. A lot of the time groups "leave it to the DM" to make sure players get equal spotlight time, to make sure the plot keeps moving, etc. These are NOT DM responsibilities, despite common lazy playstyles. The best thing you can do for your new DM is not to get in his grill about DMing, but to be a group-aware player instead of the (unfortunately common) narcissistic little snowflake player. Help keep things moving, help mediate party disputes, help give other party members times to shine. The DM adjudicates, but anyone can lead.
Have an honest talk with the new DM.
Let him or her know that you're there to help and support if they need it, that they shouldn't worry if anything seems overwhelming or complicated at first, not to worry about making mistakes, and that if they don't need anything, you're just looking forward to having fun in their game.
Have that talk with the DM, not in front of everyone.
If they do want advice/help, the three most useful things I can think of are this:
Where to Start?
What rules are worth learning well early in play? Which rules can you read up later? It cuts down the game from reading the whole book to only focusing on a few chapters, which is really helpful.
"Here's a few articles/forum/blog posts that I've found really useful."
"If you ever want help remembering how a rule works in play, just ask."
If they ask for more advice beyond this, you can start going into further details. ("Oh, yeah, against high AC opponents, these kinds of monsters are pretty good, etc.")
I feel like this might get a little redundant but I've worn my DM camp backwards and pretended I was a player in a variety of situations. These include sitting at a game I've run (sometimes several times) where I'm just sort of a codex/index for the DM in question to having never run the game but having read the rules thoroughly and helping the DM in question with parsing and application. But essentially it does come down to things that have been said:
Only spin your DM cap back forwards when they address you for the help. Having "Lawful" tendencies makes this one the most important for me personally because while not rules lawyering, I come to the table with expectations of how things work until I see the emerging style of the DM. If it helps, consider this like you and your friend are both managers in the same chain and for some reason you both work at each others' stores from time to time. Even if you have the same elevated positions, you're working in their store and they have the authority. Besides Rule 0 makes them right even if they're not.
Offer Abstract Planning Tips
Occasionally a fellow DM wants some leads when they have an idea but who can they turn to? Try to give advice that's generic but add some notes for trickier things you've noticed so that there's nothing meta going on for you but you can still be of help in significant ways. Oftentimes the brain trust (trust a key word) is the best way to pan things out for a better game as long as they know it won't come back to bite them.
If you and the DM have a powwow about how they plan on running things, you can help the other players create scratch sheets and get the messier details of their turns prepared if they are also new to the game. You don't need to be the leading character but when the DM and the PCs are scrambling it's chaos so any touch of order can bring things back to center.
I was "that guy" for a few sessions in a campaign with a new DM a few years ago, and it took a while to realize that my interjections were more annoying than helpful. Sometimes relaxing as a player is more difficult than organizing the campaign, once you sit down at the table.
Most things I would recommend have already been said, but something that helps a new DM a lot is to try and answer rules questions other players have while the DM is doing things. Also, starting the conversation about "what should we do now?" while he's preparing is more helpful than it looks-- just filling the silence takes a lot of pressure off.
I've been in this position a lot. I am a very experienced DM and, by all accounts, a dang good one, but I have a strong preference for playing and leaving the headache of world building and encounter planning to someone else. Because of this preference I usually find myself playing under a less experienced DM, especially since life forced me to move away from home and the group I had gamed with since high school (at 39 I'm now the 'old man' who's been gaming since most of my current group, mostly college kids, were in diapers, but thankfully one of them enjoys running so I haven't been forced to).
Short version: accept that they're inexperienced and going to make mistakes. Make sure they know you're willing to help and they may ask for advice, and it's ok to help then, but whatever you do DO NOT contradict them in front of the other players. If you feel the need to correct them on something do so in private and do so only very sparingly. If you're pulling them aside every other session they're going to resent you.
It does, of course, depend on their temperament. Some people just aren't secure enough to let you help them. If that's the case then your best bet is to sit back and enjoy the game.