None is very felicitous. OED has this example:
1978 Jrnl. Royal Soc. Arts 126 617/2 It will be a hundred years plus before we have a significant contribution.
...which would indicate that "One year plus of experience" would be better than putting plus before year. It does sound better.
Better still would be to avoid it altogether and write "More than a year's experience..."
Having read the question better, it appears that there is no option on word order and the query is about whether years should be possessive.
Any positive number which is not exactly one is treated as a plural, but if we were to ignore the plus and treat it as exactly one, it would be written as either
- one year's experience
- one year of experience
Using a possessive form with a plural number we would write "two years' experience", so if that is to be used, then one plus years' experience would be correct. As would using of with no possessive, as suggested.
Neither reads well.