To be honest, I wouldn't do this.

There are some buses, but you'd probably end up going on the train, and the fares over the bridge aren't especially cheap (SEK106 each way). It's an easy and impressive journey, but it does take some time.

I'd look at alternatives in Copenhagen or in Denmark. If you're willing to travel to save money, it'll be cheaper to travel on the Danish side. Plus, there are some decent rates out there in Copenhagen: AirBnB has some very decent rates in the city, for one. Plus, you often get a bike thrown in (and I'd almost class that as an essential for the city!)