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With regard to the carpet (also known as a rug), it depends on the material making it up. Generally, the longer and more frizzy the strands on the carpet, the worse off you are. My apartment has a carpet that's extremely flat, and I've never had static build up. The carpet in your picture is a bit hard to tell since you took it looking down, but it seems to be similar to the carpet in my girlfriend's apartment, which isn't quite flat but the strands are still very short (as fun as it'd be to show each other pictures of our carpets, I'm not breaking out my camera for this one ;P ). That's OK to work on. By comparison, my father's house has this awful green carpet, which is also referred to as a shag carpet. You can't walk more than five steps without generating enough static to shock yourself every time you touch something metallic. Absolutely do not do any computer activity on that rug.
As for grounding yourself, the first time I opened my computer, I used one of those anti-static wristbands. I don't really buy into it anymore. Assuming your computer has a three-prong plug (I'm pretty sure that this is standard in countries outside of the US as well), leave it plugged in, but shut off the power supply. To ground yourself, simply touch anything metal on the computer case. Before using any metal tools, touch them to a metal part of the case. That grounds them.
For laptops it gets a bit trickier. You're expected to unplug them and remove the battery - grounding is then accomplished once again by touching anything metal on the case. I get nervous about that because the computer isn't connected to a ground source, so in theory if you transferred enough energy to the case you'd still experience static among some of the components.
Enough rambling - your rug is likely fine (you can post back a picture of the rug more from the ground level if you want us to check the carpet length), just follow good grounding practices and you should be fine. Note that you should be following good grounding practices no matter where you're working.
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