With care, you can use your body as a resistor to prevent issues. Let's say you are installing the motherboard in the chassis - you touch the chassis, and at the same time you touch the bag the motherboard came in (and is still in at this point.)
Then you open the bag and get the motherboard out - which means you don't still have a hand on the chassis unless you have three hands - so before you put the motherboard in the chassis, you hold it in one hand, and touch the chassis with your other hand before bringing the two into direct contact. You do the same thing when installing memory, etc. So long as you remember to be careful, you can do the whole job without a strap, and without a problem.
OTOH, a cheap wrist strap adds a degree of insurance - it's the third hand in touch with the chassis. A proper table mat adds another. The wrist strap is cheap enough that it makes sense for building one PC. The mat is expensive enough that it probably does not - and in assembling a PC, you really are mostly installing components into a conductive chassis, so the function of a mat is not really needed, so long as you transfer components from their anti-static packaging to the chassis without laying them down on the table. The mat is of more use when you actually need to lay things on the work surface (ie, you're soldering them or otherwise "doing work on them" rather than "putting them together.")