IHATE having blood work done. My body does NOT like having needles put in it and it makes my blood pressure drop and have quite a reation, this is quite rare though. I have found that laying your head down so your head is below or at the same level as your heart decreases the reaction I get.
ask them if they can lower the chair back so you can lay flat, since nerves can account for this problem as well.
another thing that makes me squeemish is actually seeing the needle. I always ask the nurse not to let me see the needle at any time, lol or I'll be a gonner. So you always have the option to tell the nurse you dont want to see the needle 
good luck, I've never had mono, but my boyfriend did for about 4 or 5 months. I continued to kiss him and snuggle with him about 2 months after he was diagnosed ( we couldn't stand it, it was a new relationship) and I never caught it. We still don't know how he got it and I didn't, but its possible I was a carrier and didn't realize.
mono is kind of routine testing now, you should be fine!
~*Jill*~ Teacher, Advanced BSc in Psychology
"You can unlock any door as long as you have the right key". Mrs. Brisby, Secret of Nimh