Connecting 3 wire timer to 2 wire circuit
No. The white wire to the old timer is actually the load, not neutral (and technically should have some black tape over the white insulation to mark it as such).
Unfortunately, your switch box does not have a neutral wire (they were not required in switch boxes until relatively recently, 10-15 years ago).
You need to return the new timer and find one that does not require a neutral connection.