Normally, you download the apk file of the new version and install it to upgrade your old game files. The save files are not changed and can be continued without problem. Someone reported he was able to delete the game, install the new version, and still had his save files, as they're not stored in the folder of the game on android. But I can't promise that this works on all phones.

Saves have weird name and location, maybe developer try to hide them?) They stored in one file (binary format). Saves location: 

/data/data/id.renryuuascension.rpg/app_webview/Default/Local Storage/leveldb/000003.log

You need root access to copy/replace this file.