This is a follow-up question to "Is a Giant Spider rideable?"
Would the riders of a giant spider be subject to checks if the spider went on walls or on the ceiling? What checks at what DCs would be appropriate?
Well, the answer to the previous question is up to the DM. Obviously, anything that follows up from "the DM allowed it" will also be up to the DM to rule.
So, splitting your question.
I would usually say that every action needs a check, unless
I doubt anyone will be ruling it as too hard. So, the question is: is it easy enough to not require a check? - We will be back to that when discussing the DC.
The most common check for this scenario would be Athletics (Strength) or a Strength Saving Throw. It is the check used for grabbing something and hanging on it against an opposing force.
An example of the saving throw being used is given in Lost Mine of Phandelver
When a flood is activated, the characters are swept away. They can try to hold on rocks, requiring a DC 15 Saving Throw.
It heavily depends on the scenario. I have usually ruled that characters could hang themselves in objects while not moving without effort - so my following answer is consistent with that.
If the spider is moving slowly, it should be easy enough and not require a check at all. If it is moving normally or quickly, it should be more challenging to the character. The easiest translation I can think of is using the same DC you would use for climbing that wall anyway - the advantage of using the spider is that it does not slow his movement speed by half.
For characters being attacked, the closest I think we have in the rulebooks is the Concentration mechanic, which states
Taking damage. Whenever you take damage while you are concentrating on a spell, you must make a Constitution saving throw to maintain your concentration. The DC equals 10 or half the damage you take, whichever number is higher. If you take damage from multiple sources, such as an arrow and a dragon’s breath, you make a separate saving throw for each source of damage.
it would be more a Str saving throw, though, than a Con. Note that being attacked while mounted does not dismount you in any way - so, by the rules, you can as well simply ignore the attacks (for the mounting process).
For the ceiling scenario, it would probably be very hard or even impossible to stay on the spider's back without any kind of extra support.
Since riding giant spiders is not in any dnd5e rule book (as far as I can tell) we are in homebrew/DM fiat territory.
I think it reasonable that staying on a spider climbing a wall/ceiling would be difficult and would require a skill check (there are consequences (falling), it is possible, and non-trivial). I can see 2 different scenarios: riding with a saddle and riding without a saddle.
Assuming an exotic saddle, it should be designed to help the rider stay mounted even in the unusual cases of being upside-down or side-ways (since the saddle was meant for spider-riding). Thus I wouldn't require any skill check just to ride, but any difficult maneuver would require one (see last section).
RAW, a (military) saddle only gives advantage on checks to stay mounted in combat but I would argue that staying on an upside-down spider without something strapping you in would be impossible without physically holding on. Thus I would call for a Strength (Athletics) check.
If the spider goes vertical (on the wall) I would call it DC 15, and if it goes upside-down DC 20. Success means the character doesn't have to save again as long as the spider keeps doing the same thing.
If the mount or rider is forcibly moved, or attempts a particular difficult maneuver I would call for an additional saving throw to stay mounted.
In the PHB in the section on Mounted Combat: Mounting and Dismounting it says:
If an effect moves your mount against its will while you're on it, you must succeed on a DC 10 Dexterity saving throw or fall off the mount, landing prone in a space within 5 feet of it. If you're knocked prone while mounted, you must make the same saving throw.
So a Dexterity save makes sense to stay mounted here. I would keep it at DC 15 for vertical and DC 20 for upside-down, with a saddle giving advantage as normal.