I doubt it. I'm using a MacBook with integrated graphics and using a 19" external monitor with absolute no noticeable drop in performance at all.
The MacBook Air may be underpowered for it's price, but it's still not just a toy. It should be able to run Photoshop and Flash, albeit not as well as say a MBP, but obviously you're looking for something in terms of portability.
That said, the machine is probably not designed to sit on a desk most of it's life hooked up to an external monitor. If you don't need the portability of it, look elsewhere. If you really need something that thin, then it's a great device.
Apple's tech specs on the device has this to say about external monitors:
"Extended desktop and video mirroring: Simultaneously supports full native resolution on the built-in display and up to 1920 by 1200 pixels on an external display, both at millions of colors"
The resolution on a 23" ACD is exactly 1920x1200, so you should be fine. Again, don't get the machine if you don't need the portability. I'm already sick of people complaining about the MBA, and I'll have no sympathy on anyone who doesn't understand it, gets this as their primary computer and then complain when it doesn't cut it powerwise. It's a capable machine, but if you need heavy lifting in CPU power, it's not fit as a primary machine.