We just returned from Uganda. I thought it was going to be hot. It wasn't. That's because the parts we went though - Entebbe, Kampala and the West - are highlands - at elevation. It was weird behind at the equator - the temp was in the 70's. I don't remember seeing any spiders. Maybe the temp this is the reason? No guarantees though! I'm sure they are there. But, the place isn't crawling with them like in the Amazon.
My recommendation would be to go to the zoo and spend time looking at the critters behind glass. This is a technique called "flooding". Of course, you will not be cured. (I'm still afraid of swimming in the ocean.) But it is a totally safe way to reduce the hold it has on you. (Just finished my first triathlon last weekend.)
BTW, the Gorillas in Uganda were one of my life's peak experiences. To be so close to a whole family of Gorillas for a full hour was just fantastic.
My 2 cents,
Darren