I have to renew my US visa every year, but currently a round trip flight costs me about $1000. Is there any way to travel to Russia, and then 1-4 weeks later return, for less? I'm a Russian national, if it isn't obvious.
One possibility to consider is to fly into another country and then cross over to Russia by land, if time and your visa situation allows it. For example, a flight to Helsinki, Finland may sometimes be significantly cheaper than a flight to Russia, and it's easy enough to cross into St Petersburg from there. Consider Latvia and Estonia as well.
Another idea is to use a European budget airline that flies to Russia (e.g. fly into the UK, then easyJet flight to Moscow)
One of the best ways to save money on a route you flow consistently is to sign up with the airline (or airline alliance) and accrue miles. This will get you a lot of money off on your flights, or extra baggage or class upgrades. However you do have to fly frequently.
I can also recommend finding a travel agent and frequently working with them, as they can take care of the miles and get you great deals you won't find on expedia or so.
Other than that, there are rumors around when the best time to buy your airline tickets are, but basically flying is simply just expensive.
I don't think there are other ways to get there besides flying, if that is what you meant by your question.