What is the best solution to sync a mongodb instance in local server with dynamic IP (set by ISP) with a mongodb instance in public server (eg. Amazon AWS)? Can i do that from node.js ?
You can do this in a number of ways, but first to address the public/dynamic IP issue you will want to either use a hostname --> IP address mapping that you maintain (/etc/hosts or your own DNS servers) or look into one of the dynamic DNS solutions.
Once you have the changing IP address problems solved, the question is how to keep the systems in sync. The most obvious way is to have the two nodes in a replica set - if your connection is reliable enough this might work, though you will probably want to put an arbiter locally or remotely for whatever side of the connection you want to do writes on when the connection is flakey (in a 2 node set, if either node is down then they are both secondary and cannot take writes).
Another option is to use the mongo connector which lets you sync to arbitrary destinations, including another MongoDB instance.
That project will give you a pretty good idea of what you need to do (in python) to provide such a syncing service. You will need to write something similar in node.js to achieve a proper sync and essentially you will need to tail the oplog on one host and apply it to the other on a regular basis, depending on your requirements.