There are so many to choose from and most will be sufficient for you while you are staying here, however these are my suggestions, based upon what I use daily.
JED:a Japanese-English dictionary
Download a Japanese-English dictionary. JED from umibouzu.com (play store/android) is by far the best one I have come across. The user interface is smooth, there are so many categories including onomatopoeia, and the best part, it doesn’t need an Internet connection to be used!
The next must have app while you are in Japan is from Hyperdia. It is a train schedule timetable. Just punch in the station you are at and what station you want to go to and it will give you several different options based upon what time it is now. The app also has the ability to be set so you can pick a time you need to leave by and it will show you the closest possible times. But it can also be set to the time you need to arrive by and then showing you what time you must be on the train. The app has the ability to show local trains only, Shinkansen trains only, and other types of trains, based upon what mode you desire. Once you have your starting and ending train stations set and the search button is pressed, the app gives you your options on which train to take. You can then make the decision by time, train line, or cost. If your destination requires you to change trains, that is all in there for you as well. This app however requires an Internet connection.
Another great app to use is Google translate. It is very straightforward and simple to operate. If you have previously downloaded a Japanese keyboard then you can even use it when trying to figure words out, or if having trouble communicating with someone, you can hand the phone to them and let them type in what they are trying to say. This app mainly uses an Internet connection, however if you have enough space on your phone, you can download the Japanese package.
If you already have friends in Japan you might know about LINE, if not download it. It is a VoIP and chat program that works very well. Making calls in Japan with your phone from the states can become expensive, and while you have access to wi-fi here why not make free calls back home or to your friends here (both parties involved must have the LINE app, and I suggest downloading it before coming to Japan). The app itself is free, additional stickers and emoji cost, but if you don’t mind spending a little for something cool or cute, then go for it. If you truly like the app, check out the LINE stores around Tokyo.
Those 4 apps will be more than sufficient to get you around in Japan and communicating with your friends. However if you have spent anytime at all on the iTunes store or Google play, you know there are so many to choose from, so pick some at random and see if they work out for you. Enjoy!