Is there a way to figure how long can a 555 run on battery, when reset is grounded, to have no output?
A CMOS 555 like the TI LMC555 has a supply current of around 300 µA at 9v (the datasheets don't have values specific for 9v, but list a maximum supply current of 400 µA for 12v, so I'm interpolating).
Holding the device in reset adds a negligible amount of current (10 pA) which can be ignored.
A typical 9v battery has a capacity of 550 mAH. Assuming there are no other devices drawing current in the circuit (which I find unlikely), just the 555 would run for 550 mAH / 300 µA = 1833 hours, or a two and a half months. You would want to derate that value, but it is likely the 555 would run for a month.
If you can run your circuit off of 3v instead of 9v, you will get much better battery life. A Duracell AA cell has a capacity of 2100 mAH compared to the 9v 550 mAH. Putting two of these AA cells in series gives you 3v, and doesn't take much more room than the 9v battery. The LMC555 has a maximum supply current of 200 µA at 3v. 2100mAH / 200 µA = 10500 hours which is over a year! If you add a 1 mA LED, 2100mAH / 1.2 mA = 1750 hours, which is comparable to the 9v battery without the LED (1833 hours).