So my husband and I are flying AA from ORD to FCO and then later Turkish from IST to ORD. This is economy air purchased as part of a cruise fare. I've got 85,000 or so Chase ultimate rewards points. Am I dreaming that we could upgrade to business with these for an least one leg of the trip? If so, what program should we transfer the points to - AA? Or is this not even doable.

Upgrades are never guaranteed. Some airfares do no allow upgrades with ff miles/points/avios. Or for some even with cash.
Getting 2 upgrades on the same flight is a magnitude harder.
Generally you can only upgrade a flight the marketed and operated by the airline of your ffp. With some exceptions.

But Star Alliance has a cross upgrade scheme, but requires a high priced base fare and has onerous t&c's. Extremely unlikely an airfare via cruse line would be eligible. TK is in Star Alliance.
UA thread --> How to upgrade with UA miles on Star Alliance carriers Are others in different forums.

And AA also has a cross airline scheme with BA & IB. But again has onerous t&c's.
Many threads in the AA forum about upgrades.
https://www.flyertalk.com/forum/amer...advantage-733/

What are your underlying fare classes and ticket issuing carriers? That'll determine if you're able to upgrade the TK flight at all, since TK isn't a partner of Chase Ultimate Rewards (but UA, AC, and SQ are), you'd need to use the *A upgrade scheme. You should be able to find your ticket numbers quite easily and that'll tell you your ticketing airline, you can tell by the first 3 numbers. 001 is AA, 235 would be TK. For Star Alliance upgrades (TK) I don't think the issuing carrier matters, but for any potential on the AA flight, it does, as noted below. Fare classes sometimes are easy to find, sometimes not, AA.com doesn't show it, you'd have to call but it's likely clearly shown on your e-ticket you got from the travel agent.

In general Star Alliance upgrades require Y or B fares which your cruise bundle almost certainly is not. But TK does also allow M, A, H, S which are still quite high priced economy fares, but getting into the range where you might be lucky.

I can't find the rules to use BA Avios to upgrade AA flights (BA is an Ultimate Rewards partner, AA is not), but it looks to me like it must be a BA issued ticket (yours is very likely 001, meaning AA issued) and unless recently changed only on a Y fare. (again a cruise ticket is almost definitely not a Y fare). AA does offer cash upgrades quite frequently though and can be decent value, especially if you consider the value of the Chase points you're willing to use if it were possible to use them. Not sure your time before the cruise and flights, but one option you have on AA, assuming 001 ticketed, you and your partner could both sign up for AA credit cards to get the sign up bonus and a good quick chunk of AA miles. Both Citi and Barclays issue them. It'd still depend if AA has any upgrade availability or not, but wouldn't hurt to have the miles stashed just in case, it's only 25k miles each + copay if you manage to find availability.
AA Mileage Upgrades