Is it Possible to be Air Assault and Airborne?

Take a reality check.....first...go visit disabled Vets at the VA hospital....talk to them about their badges and tabs...Then go visit a VA National Cemetery....and read the headstones....is your goal to be a peacock..or in selfless and honorable service...go for it...those badges and tabs are for graduating from schools.....nothing more nothing less....remember the fallen and the disabled.....honor them...not yourself....go push a Ranger in his wheelchair chair at the VA...shake hands with WWII Vets.

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.