So first off, he should definitely not rely on magic. Anyone might have catch that via detect magic and while it's a low chance he should be wary of true sight as well. Weirder things have happened, like his birth at all for example. So every morning he should spend an hour applying a disguise the mundane way. Mechanically this will actually be similar to his spell save anyway. Actually it'll average slightly higher as it's probably a passive check that others have to use investigation or insight against, provided of course he has proficiency in Disguise kit, which there is no reason for him not to take. Regardless this means if for some reason the spell fails, he'll be fine. It also means is face will feel correct if he has to kiss the back of a palm or anything.
In addition he should always he dressed in the finest foppery, that covers the most of his body possible. This will help throw off his body shape again if for any reason his spell fails.
He may also want to adopt a reputation as a recluse or sickly. That means fewer times he'll have to appear in public and easier excuses to leave. This will also help perverse the illusion in case he doesn't do it exactly the same one day to the next as he likely wouldn't(though there's nothing in the mechanics for that)
Lastly, for spells in general Disguise Self is good, but requires frequent upkeep drawing suspicion. If he could cast seeming it would be better as with extend spell that will be 16 hours, pretty much a whole day. However that requires he's a level 9. But since he was taken care of by a powerful caster growing up, it's likely they wouldn't have abandoned him with out making sure he could take care of himself. So it's not hard to expect that he might have a stack of scrolls for seeming. Each is valued at 2,500 which is alot...but he's a prince with a dragon mother. So even if whatever wizard was taking care of him couldn't leave him with them he would still be able to get his hands on alot without too much issue. Consider in non fantasy England Richard the Lionheart was ransomed for 150,000 coins and that's in a world where the economy makes sense. It shouldn't be hard for him to raise enough to handle it. He can cast the spell as a level higher with an ability check if he thinks he'll be out for a long period of time. If he fails he can simply use disguise self as others have pointed out or try again if he really needs to be careful for some reason.