What is the code punched on the spindle? Should say something like 3S. Replace the spindle like for like, and while you're at it, replace the ball bearings for the cost of a few bucks. Parts on this should be less than $10, particularly if you can find a bike Co-op.

Replacing with a sealed cartridge bearing bottom bracket will cost you $30+ for the BB, and even more for installation. And then, you may have to settle for a spindle size that doesn't exactly match your original. Plus a 'sealed cartridge bottom bracket' may not be particularly well sealed, and will certainly have more rotating friction than a well set up cup and cone BB.

Someone competent should be able to replace the spindle in less than 30 minutes, from the time the bike gets clamped in the stand to when it comes off.