I'm probably stupid, but can I play a tab in that is in standard tuning, the same way with a guitar tuned down to C#? Will everything just sound lower? Or do I need to get a tab in that tuning?
If all the strings are tuned down three half-steps then yes, you can play the tab exactly the same and it will just sound lower — specifically, the song will be transposed 3 half-steps lower. Naturally, any other instruments would need to play lower to accompany you as well.
If you're playing Drop C# (everything down 1 half-step except the low string which is down 3) then you would need to adjust the tab, most likely by playing everything on the lowest string 2 frets higher than the tab indicates. If that doesn't work you would probably need to rewrite the tab to fit some other possible way to play the song.