Yes you can, you can run one monitor from the normal monitor port and another from a USB port but you need a special connector which makes the Air think it has a 2nd monitor attached. Here's an example on YouTube
You need something like the Matrox Dual Head 2 Go which allows you to have multiple monitors with your MacBook. The performance of USB-attached monitors tends to be worse than those connected with display cables so they would be fine for regular computer use but probably not for gaming for example.
EDIT
With the release of Thunderbolt-capable Macs, you can now drive dual external Thunderbolt displays from MacBooks which have a separate graphics card.
Unfortunately, there is not a cheap good or ideal way to do this.
The graphics card is a key integrated part of the logic board on the MacBook Pro, there really is no separation of it from it to do an upgrade, short of replacing the entire motherboard, and even then you would be limited to what was available at the time for that generation of MacBook Pro.
See that orange highlighted square, thats the NVidia GeForce pictured below:
Honestly, the Matrox route is probably the most affordable workable solution to this problem.
This MacBook Pro also lacks an ExpressCard/34 slot or Thunderbolt so you really can't go the route of an external graphics card. And even if you could you would be talking over $500 USD just to get the external PCI express chassis, for example Magma's ExpressBox 1 starts at $829 for just one external PCI Express slot, you would still need to spend another ~ $200 to buy a mac compatible graphics card.
Your last option is to maybe reevaluate if this MacBook Pro is the right machine for your needs. Would a newer model MacBook Pro with Thunderbolt solve this problem, maybe maybe not. If you don't have two Thunderbolt Displays, you are kind of still out of luck hooking up dual displays to even a Thunderbolt MacBook Pro, with out the need for expensive external adapters.
Would an iMac, Mac Mini, or Mac Pro better suit your computing needs, complimented with a basic MacBook Air or iPad of some kind for your mobility needs.
Best Answer
I have this product. All your concerns are valid. Modal dialogs and prompts are split by the program's bezel. The dock can be positioned across only one monitor, but that is the only configuration option. Fullscreen video on one monitor is not possible. Fullscreen apps span both screens, which doesn't work well. Calendar is OK. Mail is impossible, especially when composing a message. It is easy to maximize to just one window using built in hotkeys. However "true" fullscreen is usually complicated by monitor bezels.