I'd say it might. How's that for a definite answer.
Really, with brifters, the older ones could use a 1.2mm cable, but the safe bet is a 1.1 designed and labeled for SIS shifting, generally with a 4mm housing. I've not had one that wouldn't work on brifters yet. The brake cable is generally the 1.2 with a 5mm housing. There are always exceptions.
My best success rate has been with the complete sets from PBK in England. About once a month, free shipping to the colonies, and the sets are about $10-$15. Since it's not a flipper, get good cables, and if you need an ally, see if LBS has one of those $24 Shimano sets. The extra money may result in some goodwill points with the shop.
I used a Specialized MTB set on a flipped Trek. It worked, but wasn't nearly as slick as the PBK. I"m about to use a Jagwire set on a Cinelli restoration. It's pretty nice.
If/when you buy a complete set, make sure of your cable lengths before you cut them. Some have the FD cable precut a lot shorter than the RD cable, so don't assume both are 60" long or anything, or you'll cut the RD one for the FD and end up running to LBS for another.
You shouldn't change the Frisbee.
1985 Raleigh Racing USA - Coleman's Competition
1984/85 Klein Performance
1988 Centurion Dave Scott Ironman Master
1989 Centurion Dave Scott Ironman Expert
1989 Centurion Carbon R