This answer assumes you have some basic skills in Excel. And the Excel program, for that matter!
 
I'd probably do a download of my inventory, open it in Excel, and sort by Core Price. Making sure my column headings were still at the top, I'd then delete all records above my minimum price, and change the prices of the remaining records to my new minimum. That done, I'd delete all columns EXCEPT the SKU number and the core price. I'd save my 2 column spreadsheet, headed with SKU and Price headers, as a tab-delimited file. I'd go to the Inventory > Upload an Inventory File page, browse for my file in the "Upload Complete File" box, and upload. That will update everything for you in one shot.