Does disposal order prevent me from playing Rhasa after from Last Rites after already paying it cost?

Yes — if the opponent plays Disposal Order before Last Rites resolves, you can't play Rhasa.

Disposal Order is a [Reaction] spell, so it can be played while your Last Rites trigger is still sitting on the Chain. It resolves first (LIFO), recycles Rhasa out of your trash to the bottom of your deck, and when Last Rites finally resolves there's no Rhasa in the trash left to play — the instruction simply fails (same principle as FAQ #11060 for Fizz; Rule 359.3.e.7).

No — if you've already paid Rhasa's cost, it's too late. The play of Rhasa happens entirely inside Last Rites's resolution (choose unit → put on Chain → pay costs → it enters). No priority passes mid-resolution, and once Rhasa is on the battlefield it's no longer in your trash, so Disposal Order can't touch it. The only window to stop Rhasa is while the Last Rites trigger is still on the Chain — respond there.