Going from 120mm bb to 113mm or 110mm bb: Help
 

my bike had an older cup and cone bb; 68 x120 dimensions.

should i be okay going to a cartridge bb with either 68 x 113 or 110?

First determine what crank you are going to use, before trying to choose at BB spindle type and length.

already have the crankset, square spindle sugino m type off an old road bike. i had used it before, but i needed a larger chainring and the bb was getting rough. the bb will be the shimano bb-un54

What was the chainline like with the 120mm BB? Losing 7mm may be a bit of an issue if it was straight with a 120.

i cant remember. meh, ill see how it goes

There's no reason to waste money, do some research and find out what length spindle the cranks call for.

Exactly what Eric said.