Do you think this is possible to do without a micro controller?
Yes, it should be possible. A 555 timer IC is commonly used to operate servos. It produces stable pulses provided that its power supply is stable and clean. Your servo draws ~350mA when moving so it probably draws 1A+ peak. A 7805 regulator may not regulate that well so you should use a separate regulator for the servo (same as you would with an MCU).
Your servo is specified to move 60º in 0.2s so it probably needs about 0.3~0.4s to go 90º. You will need enough capacitance to hold the voltage above 5V for at least 0.4s at 350mA. The 7805 needs ~2V headroom to avoid 'dropping out', so the capacitor can discharge from 24V to 7V. To detect when power is turned off the capacitor must be isolated with a diode which drops another 0.6V, leaving ~16V that the capacitor can discharge by. C = I*t/V, so you need at least 0.35A*0.4s/16V = 8,750uF.
If you used a switching regulator ('DC/DC converter' or 'UBEC') to power the servo then less capacitance would be required because a switching regulator 'transforms' the power to higher current at lower voltage, rather than just wasting the excess. From 16V to 5V is a ratio of 3.2, so the initial capacitor discharge current would be ~120mA, rising to 350mA as the voltage dropped towards 5V.
The circuit might look something like this:-
simulate this circuit – Schematic created using CircuitLab
Relay RL1 detects when power is switched on/off to change the servo pulse timing. D1 isolates C1 from the power input so that it can drop the relay instantly when power is switched off (otherwise it would be held operated by the charge on C1). R1 and C2 form a 'snubber' which absorbs back-emf energy from the relay without slowing down its release. U3 supplies separate regulated power to the servo so it won't disturb the pulse timing (replace this with a switching regulator to reduce the amount of capacitance required in C1).
One potential problem with this circuit is that when C1 eventually discharges below 7V the 555's supply voltage will drop and its timing will drift, possibly causing the servo arm to move slightly before it completely runs out of power. One way around this might be to have two reservoir capacitors and blocking diodes, one to power the servo and a smaller one for the 555. If the servo runs out of power before the 555 then it doesn't matter if the timing drifts.
Alternatively you could use a comparator to reset the 555 when the capacitor voltage gets close to 7V, and perhaps another comparator in place of the relay to monitor the power supply voltage and switch the timing resistance.