Unclogging the cooling unit (A/C) drain line on a 2002 Mazda 626?

[sylvar]

I've already asked on the mazda626.net forums for my car, but perhaps I'll find useful advice here too.

I've got water leaking from the cooling unit into the front passenger-side footwell. I took off the glove compartment and it's clearly coming from the cooling unit.

I took it to a mechanic who said she's got too many customers to handle before Labor Day, but she'd be glad to take care of it next week. (She's the one who tipped me off that it's probably a clogged A/C drain.)

I'm not sure if this is something I can take care of myself before then. I hope so, but I'm a novice.

I took off the three hex screws holding the bottom of the cooling unit onto the top of the cooling unit, and I took off the nut that was holding the bottom of the cooling unit onto the firewall post beneath it, but it's still connected somehow. There's a nut holding the top of the cooling unit onto the firewall too, but I'm thinking it might be foolish to disconnect the entire cooling unit as I've never done more than flat tires, dead batteries, and worn-out windshield wipers. I was hoping that I'd just be able to pull the bottom off, rinse it out, and put it back on.

The other strategy I thought of was to go through the firewall from the front; I can reach the A/C drain from there, but I'm not sure whether I should try to clean it with a big fat pipe cleaner, a vacuum hose (assuming I could Apollo 13 a wet-dry vac to that half-inch pipe), or a can of compressed air (but wouldn't that just shove the crud further into the cooling unit?).

What do you think, folks? If you had a younger sibling who asked you this question, what advice would you give him/her?

And, finally, am I right in thinking that I can prevent leaks for a few days by just not using the A/C?