What a shame you have so little time in both Prague and Berlin. Can you find more time? Both deserve 5 days each.

Congratulations on the fact that you at least have ONE day. You do not elaborate on interests but simply wandering around in some places might be enough. The Brandenburg Gate, Checkpoint Char. Museum, the Holocaust Memorial, the newer architecture scattered about, Potsdamer Platz with its steel and glass, Sony Center, etc.

Don't feel the least bit guilty about this whatever you do. Depending on your interests you might be able to accomplish a great deal of sightseeing in even a short amount of time.

I was in the same situation,t hough I was supposed to have 3 nights in Berlin and because of weather my flight was cancelled, leaving me with 2 nights and 1 full day. It can be done, but it will be rushed. In the one day I was there, I visited the Jewish Museum (truly incredible, I recommend it), the Berlin Wall, Brandenburg Gates, Parliament, and even had a little time to spend at the huge mall.

with a short time you have I would skip museums and follow what Dukey mentioned
plus ( from Melia) wonder over to Gendarmenmarkt and stroll on Oranienburger Strasse

Oh, forgot about the Reichstag...definite possibility as mentioned above.

Reichstag tour has to be booked ahead of time.

Cupola visit is possible with a reservation at the restaurant.

Berlin has more canals than Venice. You could take a cruise on the River Spree. Some (all?) start across from Museum Island. There is narration and you get to see a lot. It's amazing to see bullet holes in the old buildings. http://www.viator.com/tours/Berlin/Berlin-Sightseeing-Cruise-on-the-River-Spree/d488-3080DCRUISE

This is my favorite church http://en.wikipedia.org/wiki/Kaiser_Wilhelm_Memorial_Church . It's on Kurfurstendamn which is a lovely shopping street with one of the largest department stores in the world.

Berlin wall and Checkpoint Charie are very close to each other. If you are there and want some adventure and love cars do a Trabi-Safari http://www.tripadvisor.co.uk/Attraction_Review-g187323-d1546493-Reviews-Trabi_Safari_Berlin_Day_Tours-Berlin.html
Fun! Everyone on the sidewalks will stare, take your picture and wave. These were the only cars availabe in the East. When you were born your parents signed you up and if you were lucky 18 years later your number would come up and you could buy this car.

Agree about the Holocaust Memorial. Berlin is a wonderful city. You can't go wrong.

Well, people can say ignore the museums but I can't imagine going to Berlin and missing the Pergamom - even if you only have time to see the Ishtar gate. How can you possibly ignore the opportunity to see a part of human culture that is so many thousands of years old?

One can ignore anything one has no interest in seeing.

Not your trip.

If you're looking for "must sees" then the Alter of Pergamon and the Ishtar gate certainly qualify. They're pretty close to the front of the museum so wouldn't take much time in that you don't have to weave through a bunch of exhibits to get to them. Just sayin.

There is a lot you can see and do in one day -- it will be the longest days of the years that time, or the shortest nights when it gets really dark only from 11pm-4.30am.

Even though you will have many sights in a 1 mile radius around your hotel, you could get a day pass for public transport to add flexibility and save a bit of energy.

Wandering around is a good idea. Your hotel will be in a great location to walk to the synagogue and the hipster Hackescher Markt neighborhood. Roughly that area between Oranienburger Str. And Hackescher Markt S-Bahn stations and Rosenthaler Platz subway station.
And of course the sights South of your hotel, along Unter den Linden, around Gendarmenmarkt to Postdamer Platz.

I would skip anything that requires a wait or reservation, unless you decide it's a must-see for you.

The Trabi Safari sure is a unique and probably more memorable way to explore the city than a HoHo bus.
So is the river cruise. The latter also come in short trips, so it won't eat up much time.

The best restored parts of the Berlin wall are at and around the Wall memorial at Nordbahnhof S-Bahn station. 2 stops from the Melia.

In the evening, and maybe after you got back to the hotel to drop off some bags with souvenirs or shopping, you could go west with the S-Bahn to Savignyplatz in the Western city center for a stroll along the upper part of Kurfürstendamm and dinner in one of the restaurants on and around the Northern half of Savignyplatz.

Take the subway back East to Görlitzer Bahnhof station for late night drinks in the area between that subway station and Schlesisches Tor. One of the liveliest part of town, and the "quirkiest". Grab a bottle of beer or wine and watch the sun set from Oberbaumbrücke / bridge behind the Berlin skyline.
Then take the S-bahn back from Warschauer Str. station on the other side of the river to Friedrichstr. where your hotel is located.

If you only have one day, then I would suggest going on one of the highly rated, professional walking tours on offer in Berlin. (check Trip Advisor for reviews) This allows you to see the most sights in the most efficient way. Cost is usually about 12 euro per person for a 4 hour tour. After the tour, then go to the Pergamon or Jewish Museum to finish up your day, then tour the Reichstag dome in the evening.