You may be overthinking this (or else maybe I'm underthinking it -- which would be unusual for me, when it comes to obscure and often meaningless airline and flight details...).
It's not clear to me exactly what you are hoping to accomplish here. You mention "miles" but I think you have booked a paid ($$$) ticket; you paid United, you'll be flying on an Austrian Airlines plane (if you have multiple legs, you may be on a mix of United and Austrian planes). IME most airlines do not give users a convenient way/place to enter a partner airline's FF membership number (remember, FF programs are all about loyalty, and they want you to be focused on their own program, not a partner's).
Are you just wanting to ensure that you get mileage credit for the paid flight(s), and that the credit goes to your United FF account rather than Austria's program)? If that's the case, I would just do two things:
- Make sure you are/were logged in to the United website when you book(ed) the ticket. I assume you already did that. If so, then your flight (on Austrian Airlines) should be visible when you log back in on united.com. (While you are there, note the United Airlines Confirmation Number/Code (aka PNR) for your flight -- AND the Austrian Airlines confirmation code/PNR (always have both with you when you travel or interact with either airline, always use the airline-specific number when dealing with that airline). Also, check to see if there is an Austrian Airlines flight number for your flights on Austrian metal - if so, note that, too.
- When you check in for your outbound flight, check in (or stop by) the counter and speak with a live human, give them your United FF number, and make sure it's in their system, ask them to be sure that your mileage credit will go to your United FF account rather than Austrian's (I always do this before setting foot on a partner airline on a paid flight -- once you've flown the flight, it's too late, do it at check-in with a human agent). Do the same for any subsequent flights on Austrian. Keep your boarding pass stubb(s), when you get home, double-check to be sure the mileage was properly credited.
IME you need to do this (speak with a human agent and make sure the mileage will be credited to your intended account) anytime you are flying a paid ticket on a partner airline. Would be the same for TAP, Egyptair, whatever. There may be airlines that make it easy for customers to do it themselves online, but I would always ask/double-check at the check-in counter before boarding the first flight in an itinerary.
As a side note: I think it's generally worth signing up for the FF program of any airline (even obscure regional airlines you think you'll never fly again), before you fly them (and enter your FF number on their website to get it associated with your flight). Occasionally, there may be some small (usually very small) perk for members, sometimes it's useful to have a FF number with them to access your flight on their website (IIRC, TAP uses your FF number for this), and you might even get (slightly) better service from employees. There's no downside (as long as you verify before stepping on the plane that your mileage credit is going to the program you want - once you have flown, and the mileage credit has gone somewhere, it's all but impossible to get it changed to be credited to a different program, so be sure it's pointed where you want it before you walk down that jetway).
Hope that helps. Good luck.