I am a US citizen residing abroad. I file "married filing separately." and have a non resident alien spouse. He wants to fix up an apartment we own, rent it and then put the rental income into his personal savings. Do I have to declare any or part of that income on my return? How much? Thanks

1. Even your spouse is nonresident, you can file joint return as residents. Attach W7 (ITIN application) with your tax return.

2. Yes, you must declare worldwide income on your tax return. For rent income, you will complete schedule E. You can claim credit for taxes paid in foreign country.
Read: Your U.S. Tax Return: U.S. Citizen or Resident with Foreign Income

If the apartment is in your spouse's name, then it is HIS income and, as long as you file Married Filing Separately, the rental income does NOT need to be declared on YOUR return.

That said, most rentals show a "paper" loss due to the depreciation, so you may want to consider filing jointly.

Just because you own the apartment jointly under your county's law does not mean that you share equally in its exploitation.
You are taxed in the US on YOUR income and if your husband gives you none of the profit arising from its rental you have no obligation to report it since you file MFS.