Hello. Is there a place in town that can be rented out on an hourly basis or overnight?

cheers

It's cheaper to just rent a hotel for the day. 

There are pages for those services. 

http://www.legelejlighed.dk/

https://diskretudlejning.dk/

Those are just the 2 I know of. I am sure more can be found. 

Thank you

https://www.aohostels.com/en/copenhagen/kobenhavn-norrebro/

they have quite cheap rooms especially if you use www.booking.com