Interstate Rail Compacts Advancement Act of 2021 This bill establishes a grant program and makes other changes to support interstate rail compacts with improving and enhancing intercity passenger rail transportation systems. The Department of Transportation (DOT) must award up to 10 matching grants each fiscal year to interstate rail compacts or commissions for administrative costs related to the compact. In awarding the grants, DOT must consider (1) the amount of funding from and participation of railroads and other stakeholders, (2) whether the compact seeks to restore routes formerly operated by Amtrak, and (3) other factors. Additionally, DOT must give preference to compacts that are initiating, restoring, or enhancing intercity rail passenger transportation. Further, DOT must publish on its website a list of interstate rail compacts, as well as annuallly updated information for the public about how states may establish compacts. The bill also repeals a prohibition on the use of funds made available for Amtrak to carry out interstate rail compacts.