Inherited an asp.net project - How do I update?

I'm new to ASP.NET and just inherited a web project where they asked me to update a drop down list and the way the selection is managed. I know c# so making those changes isn't a problem.

My question is, what would be the best way to make this update? How do I download the project from the server (.zip?)? And how do I upload the newest compilled version to the server?

It's extremely important that I don't take down the website for long because it's a tool for our sales team.

Any advice will be helpful. Thanks in advance.