No you can't just take off the cassette and put a single cog on to go fixed, you need fixed rear wheel. You can do SS that way however. There are plenty of cassette SS conversion kits.

Back to fixed...

A lot of times you can run 700c wheels with road tires on MTBs no problemo. Most rigid MTB forks have a hole you can use a nutted caliper on, or even drill out back fork hole for recessed. If you have sproingy fork you'll probably want to replace that. If I had a bike with a sproingy fork I'd just look for another donor bike to start a fixed project from.

If it has a cassette it's probably not old enough to have horizontal dropouts so you'll have to see what kind of magic ratios work or go eccentric somewhere. This is a killer magic ratio finder.

Brakes and cassettes can be de-rusted pretty damn easily. I only go SS if the rear derailer is fubared or missing, stuff like that. But, up to you. Fixed will take more parts and luck than SS will.