You're asking too broad a question for it to be answered any more succinctly than "it depends". It depends on where it is (temperature conditions, site orientation), how big it is (and I don't just mean how many rinks or how many seats; what's the footprint? the building massing?), what kinds of amenities it has, how its mechanical and electrical systems are set up (gas heat? electric heat? how is the refrigerant plant designed? LED lights? what's the acceptable indoor temperature?), what the utility rates are, mill rates for property tax, yadda yadda yadda.
In my experience in the city of Calgary the monthly gas, electricity and water costs alone are $10,000+ per month. Property tax depends on the ownership structure, the site location and all sorts of other factors; could be $20,000/month or more. No idea what insurance costs. Maintenance budgeting should factor thousands of dollars per annum extra to pay for future equipment replacements.
It costs a lot. That's why in my experience private ownership is rare; costs are usually borne by the taxpayers.