Went for the Ireland game last November. Actual game was moved to Aarhus, which is definitely not worth a visit, but Copenhagan itself was incredible. Somehow had more Irish bars that anywhere I've ever been. Some relatively cheap nightlife near the train station. Some of the bars were wildly expensive, but definitely value to be had if you go looking. Loved Tivoli and would highly recommend. The walk to the little mermaid is LONG, but a few nice bars with outdoor fires to stop at and warm up. You're probably a bit early for the Christmas market, but there were dozens last November, which served great food. Make sure to drink some Glogg while you are there for a winter warmer!
It was actually an "Irish" bar that we camped in before the UFC on the Saturday, 20kr a beer. Sami Hyppia strolled in to watch the Liverpool game and my mate hugely fanboyed before building the courage to approach him for a picture, which then set off a stream of photo requests. Sami must have been fuming. He was seemingly very sound and was off to the UFC too.
We stayed near train station and as you said it was ok prices there too. People smoking in bars was very weird.
We cycled a lot, including down to the Little Mermaid which was fairly unspectacular. Hired the electric scooter things a couple of times, I've never been good at balancing and went flying once. Shrugged off the shoulder pain all weekend (aided by beer) but had to visit doctor today. Dangerous things them!!
Such a pretty place to look at it, and I don't tend to appreciate pretty stuff like buildings etc. Food was very decent and with some leftover change snuck in a tattoo and an escape room on the last day.