Hello, I currently work as enterprise web application developer, I have 3 years of work experience. I studied bachelor of engineering in software systems for four years. what you're asking is relatively easy. So here are the answers to your question:
1) This would cost from 10-30 dollars a month to host it. This is an average hosting fee for a simple database with average load (requests and queries made to it).
2) Based on my understanding, you have a database, and another person has another database. you are also interested into the information they store onto their database, and want to store it in your database too. That is synchronization that is you're asking for. Now there are two ways you can do this, one is the technical way, through scripts on your server. Or through a website that interfaces your database and the person database. Then you can either choose which client info to copy to your database or you can control when to trigger the synchronization.