If you are allowed to stay in Germany, so you have a "Aufenthaltsgenehmigung" normally you can move to any city you like. Regarding documents needed, I think its best to contact one of these following institutions in your corner:
Bad Doberan/Rostocker Land e.V.
Tel: 0173/ 2310795
DRK-Kreisverband Güstrow e.V.
Haus der Familie
Tel:0172 847 466 3
Because sometimes there are special documents required in the different "Bundesländer" to get the Jobcenter or Sozialamt payed for your appartment (In case you don't earn enough money, yet). Also, they will know best how you can move away from Güstrow.
I also found this website, where buildings are shown to hosts people that don't directly find a new place to live. During the period of living there you will also get advices from them and help to start your new live in Hamburg. But you would live with other refugees there, though. : http://www.foerdernundwohnen.de/unternehmen/alle-einrichtungen-im-ueberblick.html?bezirk=&bereich=2&angebote=6&zielgruppe=
(I know its in german, Im sorry but it doesn't exist in english. But when you click "mehr" underneath the text next to the pictures, you will find contacts where you can call or email them)
Maybe an opprotunity if you can't find a flat that fast!
After you got all the documents and advises required to move, you can start your search for apartments or flat sharings as the link Jan gave to you describes it.
I wish you all the best
let us know if you have further questions and if you managed to find a flat in Hamburg :-)