Yes, you can say :

“J’ai eu sommeil pendant le cours parce que le prof était ennuyant”

It’s the same than :

“I was sleepy because …”

Many possibilities.

J’ai eu du mal à rester éveillé (parce que le prof était ennuyant).

Seems the best to me. There is the idea that you felt sleepy but you didn’t actually sleep.

J’ai eu sommeil pendant le cours parce que le prof était ennuyant.

Is grammatically correct but is strange as “avoir sommeil” suggest that you are tired more than sleepy.

J’ai eu sommeil is fine but ennuyant is no more use in that acception in France (see http://bdl.oqlf.gouv.qc.ca/bdl/gabarit_bdl.asp?id=2410 ) :

j’ai eu sommeil pendant le cours parce que le prof était ennuyeux.

or better, to emphasize the fact the teacher was really boring:

j’ai eu sommeil pendant le cours tellement le prof était ennuyeux.

If you were really about to sleep (but didn’t really), i.e. you dozed off, you might also have said :

J’ai somnolé pendant le cours tellement le prof (or le cours) était ennuyeux.