Timer Circuit Schematic

I have a schematic for a timer circuit (see link). Can I replace the battery with a capacitor?

http://www.freeinfosociety.com/electronics/schemview.php?id=2038

You could, but the capacitor would be impractically large.
The 9 V battery has roughly 100 mAh of charge, with an end-of-discharge somewhere above 7 V.
This translates to energy of about (9+7)*0.5 * 100 mAh = 800 mWh, or about 2880 J.

For a capacitor to deliver the same energy over the same voltage range:
E = 0.5 * C * (9^2 – 7^2)
C = 2 * 2880 J / (81 – 49 V^2) = 180 F

180 Farads is a pretty big capacitor, even for the electrochemical double-layer caps, aka ultracaps or supercaps.

Stick with the battery.

24 Hour Relay Activated Timer circuit