well it ain't downtown, but the metro will take you there.
Stations you will need - christianshavn for christiania and then either kongens nytorv or nørreport for the rest. kongens Nytorv is where you will get off to see Nyhavn. Its a great place in the summer, as everyone buys some carryouts and sits on the harbour side to drink and have hygge whilst watching the tourists and the swedes drink £7 beers in the bars and cafes.
Nørreport is pretty central and you can walk from there into the central town. You can think of the town as one central bit with 5 different bridges (bro) around it although there isnt any more the name has stuck.
Østerbro - the east bridge. The chelsea of copenhagen, expect to get your ankles run over by yummy mummys, pushing prams more expensive than cars in the UK, whilst sipping lattes and yabbering into iphones.Lots of cafes for ladies what lunch.
Amagerbrø. Bit more essex. Expect to see big tattoed guys in vests with fighting dogs and blond girlfriends. Amager is known as the shitty island, because it was pretty much just the towns rubbish dump for hundreds of years. This is the island your hotel is on. Although øresund, the area the hotel is in, is a newly designed town, full of some very impressive architecture.
Nørrebro. This is the most mixed neighbourhood. Although, poor compared to other areas, Some nice cool bars / cafes hidden around. Always good for a schwarma at 2 in the morning. Jægersborgade is a pretty cool place here. They cal it little Berlin. It's better during the day, full of trendy little boutiques and cafes. No bars or restaurants I don't think.
Vesterbro. The red light district. Istedgade from the main station is, ahem, lively, full of sex shops and prostitutes, drug addicts and crazies. Of course, the hipsters have picked up on this and there are some great places near here. Kødbyen, is the old meat packing district, it's where that restaurant i mentioned above patepate is. I also recommend mother in the same area for some seriously decent pizza. Jolenes is a good bar there too. It used to be in Nørrebro but they moved after neighbours complained. Saw trentemøller spinning some records in there (unanounced or even noticed) once.You can also check out vega at the end of istedgade (vega.dk) to see if there are any decent bands /club nights on when you are over.
One of the weirdest things about istedgade is the crackheads who sell/use/fight out the back of the police station. The police never seem to do anything about it. Guess they figure it's easier to keep them in one place.
Frederiksborg the rich part. Couple of nice bars, nothing to write home about, pretty staid.
if you smoke weed. Christiania is the place to go. It's no way as touristy as Amsterdam, it's not legal in DK, but "tolerated" in Christiania. I don't smoke ganja myself anymore, but if you do, you can buy prerolled or a bag. It's all in grammes. So fuck knows what that is in eighths and quarters. Roughly same price.
Food, it's expensive here, but you must try at least sild - pickled herring, flæskesteg sandwich - roast pork belly sandwich, usually served with pickled beetroot, pickles and remoulade (sort of weird mayonaise).
Drink - like london prices but a little bit more. Depends where you go of course, the touristy places around tivoli and on strøget, easily £6 for a pint of pish, avoid these places like the shitholes they are. It should be quite apparent. There's some good microbreweries around too. Nørrebro brghus has some interesting beers, all brewed in house.
Travel. Get a klipkort. You'll need a 3 zoner to travel from your hotel. unless you plan on getting a taxi everywhere. Just note, there are no ticket gates. You are expected to clip the card on the machines when you get down into the station. The fines about £80. it's very easily done particularly late at night after a few. The klipkort works on metro, buses, overland trains (s-tog), kystbanen (coastal line) and river boats. The city and surrounding areas are split into zones. you count the amount of zones you passthrough. You can buy one off tickets at the stations but these work out more expensive. Watch out for taxi drivers, some may, as soon as they realise you are not Danish try and rip you off by taking the scenic route. Although, on expenses it doesn't really matter, bit of sightseeing innit.
Anyway, it's a pretty relaxed and chilled place, Denmark, you'll enjoy it.
And don't even get me started on the women, for some reason, they love the English. although I suppose the guys are pretty good looking too if that's your bag. I like to think I have even got better looking by living here. Purely by osmosis.
If i think of anything more I'll put it down.