I have two alternative solutions for you.
1. Zip it
Make a zip archive (or RAR) of the multiple items you want to upload. Then upload only that file.
Zipping has the advantage of preserving LAST MODIFIED (folder uploads tend to not preserve that field)
2. Auto file sync tools
Alternatively, you may choose to use a 3rd party tool like FolderSync or Autosync to sync entire folders automatically and periodically in the background. This kind of gets you the usual Desktop backup behavior you might expect from Drive. Unfortunately the current version of their app won't do auto backups. Using FolderSync or Autosync will allow you to sync entire folders easily without having to tap-check each file you want to sync in the folder.
I do not recommend using the Drive app itself to upload multiple files at once (it gets very slow and heavy with 500+ items).