Can I replace a 0.1µF capacitor with 1µF in this circuit?
Seeing how it is across the 9V supply rail it is just decoupling for the power supply rail. Better designs (as most people here would be quick to point out) would have a decoupling cap local to each Vcc pin per IC, but for a simple design like this, one cap across the rail is probably fine. Using a 1uF cap is okay. Just make sure it is rated for >9V. Safest bet would be 16V caps, 10V is cutting it close.

Truthfully this design would probably work with no cap too, you might just experience weird glitches every once in awhile if the power supply blips from sudden current demand changes.