First, the assumption is that the person has some background that prompted management to make them "responsible". Classes and seminars are not the only way to learn. Being "responsible" means taking advantage of all of the opportunities to learn: books, manuals, mentors, web, forums, etc.

I have often been responsible for assessing or implementing new technology/software. The first thing I do is look for manuals or books to use as a reference and to get the big picture before trying to deal with the details.

Fortunately, you can find the answer or guidance to most anything on the web these days. There are a lot of DBA "cookbooks" out there. Figure out what is most important and start there - backup and recovery most likely. And, don't forget networking and firewall basics.

I have seen many people who simply refuse to take any initiative to learn on their own. They wait to be trained by the "company" until they are no longer a needed resource or their salary stagnates.