Grab highly recommended books, read and study like mad. Since you're playing catch-up, you're going to have to spend a lot of your own time to come up to speed.

Anything that is operational related, security, backup/restore, make sure you understand exactly what you're doing before you do it. Double check your steps before starting. Make sure you have your sources identified (Books Online, MVP blogs, forums like here) and well-researched so if something goes wrong, it's not because you haven't tried to cover every detail.

With respect to development work, look at your query plans. Find out which ones don't perform well. Run the traces to determine how the app is performing. Bring forward recommendations of what to fix, as well as how to fix it. Quantify improvement where possible (test a fix on a dev server and show the difference in query plans, for instance).

In other words, seek to at least satisfy whenever anything is asked of you. Look to excel whenever possible. You're trying to prove to them that they aren't making a mistake letting you take the reins. Therefore, you have to show you do your research, you are meticulous in your planning, you have good analysis skills, and you are working hard to make the project succeed.

That's how you prove yourself if you're green.