Drying paint.

This is one of the problems with acrylics as they do not adhere as well as enamels because of their nature.
I would use a good primer coat with some "tooth" to it to allow the acrylics to bite harder making them more resistant to lifting from masking and then don't worry about speeding up the drying time.
Let them dry 24-48 hours and then mask.

Mike

Oven Baking.. i had a thread for that one..
http://www.finescale.com/fsm/community/forum/topic.asp?TOPIC_ID=49531

tWhat type of acrylic paint do you use? acrylic paints already dry fast.. try thining it with isopropyl alcohol. if you are concerned about the paint not adhering to the surface when masking, try to prime your model first before applying the acrylic basecoat. primer makes acrylics adhere well to the surface to be painted.
remember to let the paint fully cure first before attempting to mask. 24 hrs minimum. hope this helps. Happy modeling!

QUOTE: Maybe bake the model for a short time in the oven???

I really don't think you want to try that with styrene! Just put a fan near it and it will dry pretty quickly.