Just copy the data to the Meteor MongoDB database - no reason to try to hook Meteor up to the existing database and risk overwriting things.
Use mongoexport to dump your collections individually, then mongoimport to import the files into the database named meteor in the Meteor MongoDB instance. The Meteor MongoDB instance runs on port 3002 with bind_address 127.0.0.1, and the data files are in the Meteor project subdirectory .meteor/local/db.
See the documentation if you're not familiar with import/export in MongoDB.
This will copy the database myappDatabase from a MongoDB server running on the standard port on localhost, to the Meteor app Mongo server. The database name the Meteor app uses is 'meteor'.
In the comments to danny's answer Tom Wijsman recommends patching packages/mongo-livedata/mongo_driver.js, line 21. A better place is in app/meteor/run.js, line 460. This way the environment variable is still picked up if present, such as when running Meteor on Heroku. Just change the default hardcoded mongodb://127.0.0.1 to the location of your MongoDB server.
Mup uses Docker, and will "link" your 2 containers, thus hosting both the app and mongo on the same VM (server). Your mongoDB shouldn't be accessible from the public IP for security reasons.