You could use the GPS in your cell phone to get the height of the barometer above sea level while it's on the roof, then put it on the ground and get a second reading, then just subtract the two.
You could throw the barometer over the building and then calculate the trajectory and the time it took it to land on the other side, but the math required to do that is way too complicated for me to figure out.
So you don't risk environmental contamination from the mercury inside the column (inherent in many of these suggestions), then go down to the local building department and pull the blueprints for the building.
Buy a cheap laser or for that matter any flashlamp with a decent beam. Move some measured distance from the base of the building. Point the laser at the top of the building and measure the angle that the laser/flashlamp is tilted. A simple trig calculation, using the length of the laser and the distance it is tilted, the accuracy can be increased by intercepting the beam a couple of feet from the ground. The tangent of the angle and the distance from the building will allow its height to be calculated.
Best done at night but be careful to avoid being hit by a barometer that some idiot has dropped from the top of the building
Max, you may want to lookup one of the Communications Society event I chaired:
Nov 2012: Technologies for Location Determination in Indoor and Urban Environments.
The speaker from NextNav (formerly @Trimble Navigation) does allude to atmospheric pressure measurement as one of the reliable ways (need 10Pa resolution). This can be accomplished by MEMS pressure sensors in a smart phone BUT it needs to be quite precise and must correlate to a reference system.
Max, I would like to add another: have the students measure the reading on ground floor, and then climb two more floors, take readings in each and get the delta in pressure between floors. Count the number of storeys in the building and use USGS data charts to get the exact height. There will be some minor inaccuracy in this since the pressure drop vs. elevation curve is not linear, quadratic.