How to get 173rd without an airborne contract?

You absolutely cannot get it without going to Airborne School, and even then, you probably won't get it as your first assignment. You could ask for Germany, which is within driving distance. But, let's be realistic and plan around that. You'll probably get a BCT here in the states, and your top three isn't a choice, it's a list that if they have an available opening and it matches with your top choices, they will slot you for. They're not slotting non airborne Infantry into the 173rd unless they've absolutely run out of airborne personnel, in which case they'd send you to Airborne School out of OSUT.
I'm surprised that you were not able to get airborne school in your contract, but they recruit out of OSUT.
Now, assuming you aren't able to get one of those Airborne slots out of OSUT, once you're at your first duty station and eligible to reenlist you can reenlist for Airborne and you'll PCS to an Airborne unit at Bragg, Italy, or Alaska (sometimes Polk). If that's not the Airborne unit you want, when it's time to reenlist again, you can choose a different location.
While someone will undoubtedly tell you that they've seen NAPs in airborne units, that was mostly during the surge when there wasn't time to send people to school, it's very rare to see, and it's not something you can request.

Talk to the Airborne or SF Liaison at Fort Benning

Well to be honest with you I don’t know but when I was there it was best time in my life