We can also assume that the netbook is pre-installed with Windows 7 Home Premium.
No you cannot assume it is pre-installed with Windows & Home Premium. It will most likely be pre-installed with Windows 7 Starter.
But that shouldn't prevent it from running Photoshop.
It should run Photoshop, but since it doesn't technically meet the system requirements in all respects, it might just give you an error message and refuse to run (not sure how picky Adobe are about such things).
Looking at the requirements & specifications from their respective websites (see below), you might have a problem with the display resolution, since the EEE PC has a low resolution display. That said, if Photoshop refuses to open on a small display, or is too hard to use, you could probably use an external display.
Bear in mind you'll be running at the bare minimum specs in terms of RAM, so you won't be able to run anything else, nor open especially large images (in terms of size/number of layers/etc) without your system crawling to a stand-still as it pages in/out of memory (you may want to consider upgrading to 2GB RAM).
Also you might need an external CD/DVD drive to install it.
Some of the System Requirements for Photoshop CS5 (from the Adobe website):
Some of the System Specifications for the 1005P (from the Asus EEE PC Website):
Yes, it can run them. The question is how well it runs them. And here I think you can expect it do reasonably well for most non-professional use. You might find that some advanced features or effects take a longer time to apply or that it might be slower with high-resolution pictures from a nicer dslr camera. Perhaps slower startup time as well. But I would expect the product to be usable.
In my experience, a netbook processor compares favorably with a Pentium D or even a very early Core 2 processor (at a fraction of the cost and power use, of course), and we all thought those were fast not that long ago. If you get get a netbook with upgraded memory, a decent hard drive, and the nvidia ion graphics chip or similar, that little system can pack quite a punch.