I see some objective cards are things like “spend x resources” “build x amount of something “

Do they have to be all in the same action or can it be over a number of actions or turns to score them?

The "Spend X Resources" ones are worded, at least in TI3, "I now spend X Resources" and would thus mean you'd have to save planets with a combined resource value of X until the Status phase and Exhaust them at that time.

The construct ones I can't help with since I don't have TI4 yet...

Yeah, all of the public objectives are things that you must do during the "score public objectives" segment of the Status Phase. For example, if it says "Spend 8 Resources," it doesn't matter how many resources you spent during the Action Phase -- none of it counts toward that objective. You have to spend 8 Resources right now, during that part of the Status Phase.