What should I clip an anti-static wrist strap to given the situation outlined in more details?
An earth ground, like a water pipe or metal ductwork is best, but you can just clip the strap to the chassis if it’s all that’s convenient. This being said, we’re moving out of static season, so a few common precautions are all that’s needed to prevent the buildup of static in the first place. Wear cotton clothes and remove your shoes before working on the computer. Memory chips and the CPU are the most vulnerable parts you’ll handle when building a computer from scratch, but you want to take care when removing any component from its anti-static sleeve. You can pick up an anti-static mat if you need a place to put down a component while you are working on it. Other than that, the risk is rather low; you’re more likely to damage a component by dropping it or bending a pin than you are to damage it through static discharge.