Well Sölden is not exactly an easy place to get to. From Innsbruck you will need to take a train and bus to get there, the cost is about 15 Euro. Use either Bahn.de or SBB.ch to look up train and bus times the OBB site is not very friendly - just use it to buy actual tickets.
Booking.com suggest that it is way too late to start planning now as there is nothing at all in the budget range. In fact all of the prices there seem to be very crazy. Engelberg in Switzerland is very popular with the Swiss and just checking it I see plenty of options for around 500 francs. There is nothing even remotely close to that in Sölden. Is there some special reason why you need to be there?
Generally speaking, you are going over the worst period of the year. The first and second of January are holidays, so people only need four days to get two weeks holidays. So naturally heading off after the 25th for a few days skiing is popular. And since Austria is considered one of the cheaper countries there is a big influx and of course the prices go up.
From my point of view, if Engleberg is still available at about 500 francs for a week, then there must be several other places in France/Germany/Austria/Italy available for less. As long as you are willing to consider other resorts and do a bit of research I'd say you are likely to find something to meet your budget.
One bit of advice I'd give you is to choose somewhere with a train station. In the middle of winter all of these places are subject to avalanches etc... which means you many not be able to get out of there when you had planned to. Usual places with a station open up quickly after such incidents, where as those relying on road traffic many not.