The federal funds pay for it through the indirect costs funding its maintenance most likely... I suspect that represents a significant portion of the total cost of the building.
Universities are only allowed to include indirect expenses that are related to the research in question, not all expenses incurred by the university (such as a student center, dorm, stadium, etc.).
Well many buildings that are donated have research conducted within them. Not always a lot, but a large building is going to have plenty of uses. Eg stadiums often have medical research that is being conducted within them.
Then that portion of that building could be counted towards indirect costs for that research, and the university is better off than if they had paid for the building as well as paying for the upkeep and those cost savings can be passed along to taxpayers.
Your imagined scenario doesn't apply to this very real example in the article in question though.
More often than not, the building is unnecessary. I'm sure the University s fine having a fancy new building, especially if the upkeep is offset by the American taxpayer.