This app was mentioned in 4 comments, with an average of 2.25 upvotes
I do exactly this, I keep my calibre database on dropbox and use this app https://play.google.com/store/apps/details?id=com.litlcodestudios.calibre_go To connect the database. Works fine for me.
The icon is exactly same as Calibre Go; other than the hue being changed.
Thanks for the tip. On that page, I also found Calibre-Go, which does the same thing for free apparently.