There are no tools available which can give you accurate data since neither Apple nor Google give you keyword data in their consoles. However, there are a few tools which can give you guesstimates derived through different ways. Apple has recently added "Apple Analytics" that provides data on device types and referring sites. They are more likely to add keyword data in future. My suggest is to check out two tools which may useful for you App Annie and Sensor Tower,

There are few online tools which allows you to analyze competitor ad space. I would suggest 2 tools SEMrush.com and Adgooroo.com.Both tools will provide very specific information about their organic search, paid search, backlinks, display advertising, top keywords, and main competitors. There are various way you can promote with small budget but again it depends on various factors which depend on what you want to get out of it. If you are looking to increase your brand awareness than a quick solution is to have display ads on Facebook and Adwords with targeted segment. If you are interested in discussing more about it than feel free to book for a call to have more personalized solution.