I can only speak for DNN not for all CMS, but considering the basic differences of how a static html-site vs. a CMS are build up i would say: no, there is no (semi-)automatic way of upgrading.
After you made familiar yourself with DNN and you found a hoster running your dnn-installation, there are at least following four steps
First you would need to design the skin for dnn to have a look-a-like your old site
This point could be hard, depending on how complex you designed the old layout, but could be easy as well.
Second you should determine what kind of content do you want to provide. Only text? Much more easy then text, video, sounds, documents, .....
You should know what is online today...so quite easy.
Third you could create an empty page structure reflecting the structure in the existing site
Well, depending on the old structure could take from a few mins to a few hours
Forth: transfer your "old" content part by part into the new system.
OK, this sounds like a bunch full of work. And this is true. BUT the great advantage: changing content of any kind can now be done without hacking in some html-files and can be done even by your secretary.
Kai