if the vacuum leaks are really bad, then yes, you would have problems right at start up. however, if they are just cracked, then it can take a while for the heat to build up, expand the cracks and leak.
It could also be something dumb, like a vacuum line used in an emissions system that doesn't get activated until certain conditions.
or, it could just take that long for the ecu to change the fuel map enough for the issue to really come through.
I was working on a car a while back that had that same problem. But it was tough to diagnose because the girl that was driving it, only had a 4 mile commute. The car was barely warmed up at that point. So the ecu was still in the "engine is cold" fuel map. But after driving much further, and the ecu changed the fuel map because the engine was hot, it would run like total balls.
The 300zx I just got done putting together ran great, until you were cruising on the interstate, then it would run like shit. It was because on the the vacuum lines related to the egr valve was rotted and the egr valve system was only activated during light load cruising.
at the very least, you want to rule out the possibility of a vacuum leak.

the other thing I would look at is the fuel system.
I would want to know if an injector was taking a shit on me once the resistance built up.
same thing with the fuel pump.

I mentioned the coolant temp sensor because it can cause all kinds of issues (with nissans at least, some cars don't use the CTS so much, I'm not sure about toyotas) Given the amount of drive time it takes for the issue to really happen, the coolant temp could explain it.

and to reset the ECU.
Unless you find the exact time, the general rule of thumb is at least an hour.
and/or, disconnect the terminals, and then touch/hold the terminals together, that will discharge any stored voltage.
usually reseting the ecu isn't a big deal with OBD1 cars though.