Buy a Copenhagen card. Go early. as you need two days before the cruise. The Copenhagen card gives you lots of free admissions, a canal boat ride, admission to Tivoli (not the rides), and unlimited mass transit. The town is compact. The Little Mermaid is, well, little. You have to go, but ten minutes does it. Rosenberg Slott and Amalieborg Slott are interesting. Look at the crown jewels. The guards are friendly; these are not Beefeaters.
Skip the Carlsberg Brewery. Not worth the time in a tight schedule.
Go to the Resistance Museum, but it is not a fun outing. The tour takes one hour. The guide is excellent.
Do not take an organized tour. Buy an Oslo Card. You can order in advance or buy it there. Admission to most museums and lots of free mass transit. After getting the card, take the Bygdoy ferry to Bygdoy peninsula. You can see the Viking, Fram, Naval and Kon-Tiki Museums. If so motivated, see Quisling;s former house, which is now the Holocaust Center.
You can walk from the Viking Museum to the other museums, but there is a bus, free with the Oslo card.
After you have had your fill of museums (Fram is the best), take two buses (it's really easy, and when you get your Oslo Card you get the bus map) to Vigeland Park. 101 sculpture's, including the Obelisk and Angry Boy.
From Vigeland, take a tram (still free) to Central Station. Cross the street and walk down the main shopping street, Karl Johan's Gate if I recall correctly, past shops, restaurants, a church or two, parliament, ending in the Royal Palace. Watch the guards, then turn back to the harbor and see the Nobel Peace Museum and the City Hall. Then back to the ship; you can wander the outside of Akershus Fortress. The tour of the fortress (also free with the card) is only available within a narrow window. The Resistance Museum is on the grounds with longer hours.
To get to Berlin you have to take a train three hours. The train is not air conditioned. However, if you want to see Berlin, this is pretty much the only way to do it from a cruise.
You need to buy a tour from the cruise or an independent operator. Berlin on your own in six hours is tough. See the wall, Brandenburg Gate, Holocaust Memorial, Checkpoint Charlie. Understand most everything has been built since 1945 as the city was devasted by WWII.
We used Alla Tours. Half the price of the ship tours, and it includes more.
Peterhof is outstanding; the fountains are on a schedule, so only go when the fountains will be on. Make sure you see the trick fountains.
Catherine's Palace demonstrates the disconnect between the nobility and the peasants. Ostentatious does not begin to describe it.
Don't miss the Church on the Spilled Blood, especially the memrial on the spot of Cazr Alexander's assasination. Don't miss St. Isaac's Cathedral, which is a huge Russian Orthodox Church. Don;t mkiss Peter and Paul fortress, especially where the Czars are buried.
Ships dock here for only a short time. You need two days. In the one short day, take a tour or the Hop On Hop Off boat (if it is running as early as you get in) from the cruise dock to the Vasa Museum. From the Vasa Museum, take the boat to Gamle Stan and wander the old town, see the palace, crown jewels, and royal coaches. Take the hop on hopp off boat the rest of the way around the harbor back to the ship.
Skip the ice bar. We've been, you don't need to go. For the same effect. pour vodka into a glass and put it into your freezer for an hour. Then drink it while staring at an ice cube.