Picking a subset with the closest average to the original set

I don't think Solver would do (natural approach with 1000 binary variables is more than Solver allows).
Systematic solution with examining all possible 250 elements subsets of 1000 elements ... wow that's a huge number.

May be ... for "normal data" you will probably get very good results by:
1) randomizing sequence of your numbers in column A
- for instance =RAND() in B1 and copy down then select from top-right to bottom-left (B1:A1000) and sort (asc or desc - does not matter)
2) first 250 shall have pretty close average to 1000.