I wouldn't recommend permatex. Instead, you should rebuild the petcocks too! :) I've been happy with the K&L rebuild kit. You don't need to replace the filters, but which ever kit you get, make sure it has the vacuum valve diaphragm. In my experience those leak over time as they age, which puts extra fuel into the vacuum system and makes the bike run too rich. Replace the vacuum lines too. Oh, and while I'm at it, replace the manifold boots (that connect the carbs to the engine) and their o-ring/gaskets. I recommend official Yamaha parts for those. As you'll soon discover, getting those carbs back on is huge hassle and you're not going to want to take them off again. Ever. Which reminds me, replace the float bowl screws with stainless steel allen heads. It's a bit tricky, but with allen heads and ball drivers it is possible to remove the floats bowls without pulling the carbs.
The diaphragms, vacuum valves and carb manifold boots could all be contributing to your idle problem, so you should deal with them before trying to tune or it will never run right.
As to the tank, if it's just a little cruddy you can probably just put some fresh gas in, swirl it around, let it soak, drain and repeat. If it is heavily rusted...well, there are lot of products and opinions about how to deal with that. I used the vinegar method on an old xv920 tank, which worked okay for me. What ever you do, here are a few tips: 1) Any chemical will etch the protective coating off the metal inside making it prone to flash rust. If it's already heavily rusted, that may be the lesser of evils. 2) Do not use loose hardware/screws/nuts/etc. The virago has a lip around the fuel inlet making it hard or impossible to get them all out. Instead use a length of chain secured so it doesn't fall into the tank completely.
Like templeofdank has mentioned, Under $50 is going to be tough. If you are willing to save your pennies though, this unit is a really good option.. I have a 1984 XV1000 and am currently using a GPS app on my phone and a decent handle bar mount. I also have a Veypor (a different unit, not the same as the one in my link) which I am currently integrating into a leather holder.
(edit: a bunch of things)