You can download an application named Budapest Go from Apple Store and Google Play Store. Within this application, you can purchase single tickets, day passes, and also monthly tickets. It is easy to use and it works for all sorts of passes and tickets, making it much easier than buying tickets at different stations or if you stand at a station without any sort of ticket machine.
You get immediate access to the tickets after purchasing them! More information about public transportation in Budapest and also a metro map can be found at the following sites.