Is it Possible to be Air Assault and Airborne?

Yes to all the above.
Enlist 11X Infantry. Ask for option 40, if that's not available ask for Airborne. In airborne school they will take Ranger volunteers. Air assault is just a two week class you'll get at some time in your career.

"Is it possible to Be Air Assault and Airborne?"
My Enlisted Records Brief (ERB) says yes

But not sure that's the best answer, MOS, duty position, duty location, timing, Luck all have some say in what schools you are afforded.

If a 20 something came in on a 4 year contract, its likely they would not be afforded the opportunity to attend both of those training schools.
If a 20 something came in to service in 1985 as an 11X and stayed 28 years 7 months, the possibilities are likely they would get most of the training they wanted. Not all, but most.

Kudos for wanting to join. Just make sure you're joining for the right reasons. It may not be your intent, but the way your statement is worded makes it sound like you're just badge hunting to brag... We join to fight, protect and defend.. not for glory and how many neat little gadgets we can put on our uniform.