Well, the first movie is going to be set entirely in World War Two anyway, so that part isn't an issue. As for any sequels set in the present, there are options:
1) He starts off as an old guy and then uses Cap's DNA to remake himself as a super soldier.
2) After Cap goes missing, his obsession with him causes him to search for him endlessly, not believing he's really dead, and when he can't find him, he has himself put in cryo to ensure that he'll be alive and well when Cap resurfaces.
3) Go with what was said in the original comics: Through sheer coincidence, he was also stuck in suspended animation through a craaaaazy accident.
The last one kind of sucks, but you get my point. There are options besides completely gutting the character.