At first the high may last you for a long while, but like with any drug, your body can and will become used to it. When that happens, you're going to need more and more of it to get the desired effect, and yes, that may or may not become more expensive. How much it's going to cost will depend on a lot. If it's legal and regulated for medicinal and/or recreational use, you could go to a dispensary and probably buy something pre-rolled for $5. But the price goes up depending on what kind of weed you want, how much you want, and if you're buying it off the street because it's not legal where you are, I'd imagine that the options would be a bit expensive.
For reference I pay a little over $30 for an ounce of weed, it's legal where I'm living right now and I have a card. I've been using it to manage my mood with bipolar. Two ounces can last me for three or four weeks. I am not sure what it would do for social anxiety. I still feel just as socially anxious when I'm high and around people but I'm more worried about my mood than I am about my anxiety. I've actually never heard of anyone using it for social anxiety but it does tend to make you either relax or fill you with energy, and theoretically those things could help.
If you are going to try it then I suggest only doing a small amount of it in a safe and controlled environment first. Weed is not the same for everyone and some people do not react well to it. You also don't have to smoke it, you can take edibles which tend to last a lot longer than smoking does for me. But you need to be extra careful with those too, because they're stronger and it's harder to control the high.
Weed can be a crutch in a transitionary period or in emergency situations, but I think it may be better to find other ways to manage social anxiety. But I'm sure you already know that, so I don't want to turn this into a lecture. I just don't want you to become addicted or reliant on weed to the point where you have to use to talk to others. Weed can be helpful in a lot of ways but it's very easy for us to become dependent on it and addicted to it, just like with all the other drugs. I don't blame you for not trusting prescription drugs, but weed has its side effects as well, despite being the safer option.