How to make 555 timer output fall
You can eliminate the switch and R1, and  connect the voltage source directly to the node where they were connected, at the left end of the capacitor. C1 and R2 will still function as a high-pass filter in the manner you need. Just make sure that their time constant is shorter than the timer's period.

It might be a good idea to connect a diode in parallel with R2 in order to improve the recovery time &mdash; anode to C1, cathode to Vcc.