Yesterday, I was turning in the garage, temp. about 90, and the lathe
suddenly slowed way down. My first thought was a brown-out, but
everything else ran to proper speed.

If I replace the motor, should I replace the capacitor as well?

Joe,

I think I'd replace just the capacitor. I don't believe it is a starting
capacitor, I believe it is actually a "Run" capacitor. Without it the
motor has very little power. And...replacing it is the least expensive
alternative. If that fails, then replace the motor.