How to add a feature like in Instagram...

You would need a web server for this. It's hard to estimate costs because it would all depend on how many users you had, what the app was doing and how often people woud be doing it and how redundant the service would have to be.

You could easily start off with a simple web hosting package offering unlimited hosting, storage, bandwidth etc etc. However in the future you may want to upgrade to more customisable services based on either a VPS (maybe an amazon EC2 instance) or a custom dedicated server.

The one option I personally use in my Dark Legacy App is using Google App Engine. The service is very redundant and offers great scalability, they offer a reasonable free package which can easily be upgraded in the future.

One thing to really take into account is the programming languages your developer can use.

Cost wise, Google App Engine will start at free and increase depending on users and the content you are serving.

Standard hosting packages (with unlimited storage) can start anywhere from around £10 p/m.

VPS servers can start at around £15 p/m and Dedicated servers from around £50 p/m

Hope this helps!