This is the concept I was thinking of (though the oscillator counter is the 4060, not 4020... )
Concept:
The two gates at the top left form a set-reset latch. In the idle state, the output of the lower gate is high, holding the 4060 in reset. From the datasheet, that may also tristate the oscillator driver, so current next to zero overall.
Pressing the start button sets the latch upper gate output high & lower one low, allowing the 4060 to start counting.
The Q outputs will count up, Q4 first going high after 8 osc cycles, Q10 after 512, through to Q14 going high after 4096 cycles.
20 minutes is 1200 seconds, so with Q14 connected to R4 and the oscillator at around 3.4Hz, it should give a 20 minute delay before Q1 is switched on, pulling 4093 pin 9 low.
The lower two 4093 gates form a monostable multivibrator.
When pin 9 goes low, pin 10 goes high and pins 12 & 13 are pulsed high by C4.
In turn pin 11 goes low, turning on the solenoid driver and pulling pin 8 low, holding the monostable active.
Once C4 discharges through the preset pot - which controls the duration of the solenoid pulse - the monostable resets.
When the output at pin 11 goes low, it also resets the SR latch, resetting the 4060 until the button is pressed again.
Once the monostable has timed out, everything goes back to its low power state soon afterwards.
[IF I've not made any mistrakes!!!]