I would say "are you up to going out?", i.e. do you have the energy, are you interested in going out?

Yes you can say the above phrase. You might also say something like "Are you ready to head out?" Or "Are you awake to go out?" Another phrase with a different meaning is "Are you up for going out?" This means are you "in the mood" for going out?

You need a gerund. "Are you up to going out?" or "Do you feel up to going out?" (ie. are you well enough?)