"One plus years experience in..."

Solution 1:

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.

Solution 2:

"One plus years" reads terrible. If the term is over a year but less than 2 years, I'd rather say something like

experience of over a year

or

over one year of experience

or similar to yours but meaningful-

1+ years of experience.

It is also good enough to write exact term in years and months since you mentioned resumés, like

An experience of one year and four months.

If it's over 1 year but less than 13 months, it's better to say

one year.