Does the schematic have to be exactly the same as the one you posted? It would be simpler to wire up to the Arduino if the switches connected digital pins to ground rather than 5V. Also there doesn't really need to be a time reset button, the Arduino could work this out. The Arduino could have 1 button:
First press of button sends power to relay and starts the timer
Arduino keeps checking relay contact until it closes and stops timer when closed.
Arduino displays time in ms on serial monitor of Arduino IDE, or if you prefer it could display it on a simple display like 16x2 LCD.
Next press of button turns off power to relay.
Then Arduino goes back to the top of this list and waits for the next press of the switch to start the process over again.
I don't have any code to share, I'd have to write it, and I don't really want to do that at the moment.
Do you have any experience in writing simple Arduino code, e.g. just to turn a relay on?
Did you drawn the schematic? I'm just asking to check if you know how to do the wiring for the button and the relay.