In article , Rookie-at-home wrote:
I've just bought a new (over 40yrs old) home. It doesn't have A/C and
it becomes very warm during the day (+85F)

I've installed screened windows & doors (to allow for circulation) but
it still becomes very warm. I think the attic (crawl space) is
trapping heat. I don't have any exterior ceiling ventilation except
for soffit vents.

If not this, what else could be causing the heat to build up? How
best to fix?

In addition to the good suggestions that you've already received, I'd like to
add this: given the age of the home, it *may* be due for a new roof some time
soon. Using white shingles, instead of dark gray or black, can lower attic
temperatures by as much as 30 degrees F.