Even if you have no intention of learning the language you should learn the alphabet. Of the entire Russian language the easiest thing to learn is the alphabet. It took me a while to learn how to correctly pronounce some letters, but just having the basic knowledge was so helpful when I needed to read street names. Plus, when you can begin to read English words written in Cyrillic letters you'll think your fluent. Also, many of these apps are written in Russian, and even if you set them to English some of the words are still in Russian.
Memrise: This was the best app for learning the alphabet. It gives you the letter and ways to visualize it in your head.
SpeakEasy Russian*: This app gives you basic Russian phrases for free. You can pay money for more, but I just used it to learn how to say phrases like, "I don't speak Russian" and "Hello." One thing I like about this is that there is a recording that you can listen to for pronunciation, and they even have an option for listening to the recording slowly.
Yandex.Translate*: Unlike Google Translate, this translator can be used offline as long as you download it ahead of time. It is also made by a Russian company so I trust the translations a little more than Google Translate. The only downside is that you can't take pictures of text like you can with Google Translate.
2Gis*: If you have to download one app make it this one! It was a lifesaver, and the best part? It can be used offline! This app will tell you how to get from point A to point B. You can use "my location" and your destination and it will tell you what bus, tram, or metro to take! The cities are written in Russian and the bus stops will come up in Russian as well but if you know how to sound out words you'll be fine. Also, if you type in a destination using Latin letters it will recognize it, but the destination will be written in Russian. You can also follow your location on the app (even when you are offline) to make sure you get off at the right place!
Rutaxi: Unfortunately, you need to have a Russian phone number for this app, but if you do or can get one you should get this app. It gives you taxi rates at a much lower price. Plus, it tells you right then how much your fare will be so you don't need to haggle with the driver. There is an option to let the drivers know you don't speak Russian and the confirmation text and phone call will be in English. Sometimes the drivers ignore the note that you don't speak Russian and will call you. I just tell them I don't speak Russian.
Умный транспорт: This app is great. It tells you exactly where the trams and buses are. Not all buses are shown, but it's still helpful. You need internet for the app, but it might be helpful if you are in your hostel or a cafe to pull it out and see when you need to leave.
Viber/Whatsapp: Both of these messengers are great because they allow you to keep up with friends and family abroad. You'll be able to send messages and pictures to friends and family even without a data phone. As long as you have wifi you can connect! Phone plans in Europe don't come with unlimited texting like they do in the States, so most people you meet will have one or the other. Both apps also give you the option to make free calls to other users. It's great if you meet someone while traveling that is willing to show you around (Russians are amazing when it comes to showing you their cities). I used these apps in Mexico, Southeast Asia, Russia, and Spain to stayed connected and keep loved ones from worrying! :)
Instagram: This is my all-time favorite app. I'm a little biased though because I like to take pictures. However, it's the perfect opportunity to show off your trip. Also, Instagram is probably the second most popular app in Russia behind VKontakte (Russia's Facebook). The only problem with using it in Russia is that you'll start to have a lot of businesses follow you. I tend to block these accounts because I don't like them, but maybe that won't be a problem for you.
I hope that you find these apps useful on your trip to Russia. Please feel let me know how they worked for you!
iOS: Settings > General > Keyboard > Keyboards > Add New Keyboard... > Select 'Russian'
Android: Settings > Languages & input > Google Keyboard > Languages > Select 'Russian'
All apps are available in Google Play and Apple App stores.