Relax and take small steps. (Score 1)

Relax it has survived so far, so likely it will continue as long as you don't make huge changes without a back out plan.

Get a security scanner software or pay someone to audit the external facing servers, that helps build confidence or scare you silly. Most things found in that kind of an audit are fairly easy to fix and low risk (patching software, limiting unneeded services and such).

Second get everything into some form of revision control that is possible. Source, images, web pages.

Back up everything into tarballs or zips so that if you make a change you can undo it quick if it goes poorly.

When it comes time to make changes do a small one first, like updating the copyright at the bottom of the page, re-deploy everything, test as best as you can. This is a confidence building measure then advance on to larger changes ...

Make sure your boss understands the current state, so it positively, since after all they hired the person before you ...