There are three options. I would suggest option #2 for now.

Vlookup Approach

If you're an excel pro and you're most comfortable with that, then you can use a vlookup. You would just need to use Apsona to export instead of import.

If you export a spreadsheet with the RE Import ID and the Salesforce ID, then you can vlookup the Salesforce ID in your original spreadsheet.

Native Apsona Approach 

If you prefer that Apsona does the work for you, you can use a little trick :) 

1. Use your spreadsheet with RE IDs as if you were importing it
2. Click on the Contact tab
3. Select import/update from the Tools dropdown
4. Upload your file 
5. Set Duplicates to Reject
6. Use RE Import ID as your Match fields (you'll select Create New... and then pick that field)
8. Click Next
9. In the Actions dropdown, select Reset Field Mapping (since we're not going to import anything actually)
10. Just map the RE Import ID field (Or RE Constituent ID, I'm not sure which one you're using) and Last Name since it's a required field
11. Click Next (Omit erroneous records is fine)
12. Don't click the Import button, rather, just use the Download match results link
14. You'll get your spreadsheet back with a new column of Contact ID's called Match Record ID
15. Always double check your results, looking for which ones didn't match and considering why. 
16. As a best practice, I copy and past this column and add it back to my original spreadsheet so that I have the ID's alongside the original data in the proper encoding and formatting.

DemandTools Approach
The above approaches are good for matching based on strict criteria such as an import ID. However, if you want a more robust matching for less strict scenarios (Contact matching or when there may be multiple matches and you have to decide between two options) then you will want to use Demand Tools. Here's the demo video of a very useful tool called FindID's.