First of all, Peter is adorable, welcoming and available for any questions. He even bought us a hairdryer before our arrival.
Then, his apartment is charming in a very calm area. 2 bedrooms (one is his daughter's room with single bed but absolutly OK for an adult, the over room with a double bed). The kitchen is fully equiped. And the bathroom has a enclosed shower (which is quite rare in Copenhaguen as I've seen in the over ads).
About area Osterbro and transports : don't worry, the center is close! You don't need metro, the bus station (n°1A) is 5 min walk and then 10 min by bus to the city center. Train station Nordhavn is 10/15 min walk, 10 min by train to the central station and from Nordhavn there are trains to Hillerod (Go to the beautiful Fredericksburg Palace!). Osterport station is 20/15 min walk or 5 min by bus for trains to Elseneur (Hamlet castle) and Louisiana (great MoMA overlooking the sea). Very nice square at the end of the street with cafés and restaurants (breakfeast, brunch, diner...). Plenty of shops on the main street nearby the apartment.
In a nutshell, don't hesitate! go to Peter's apartment!