The current drain of 1µA is the average drain of the circuit.
A capacitor over the oscillators supply terminals is needed in your case.
The formula for capacitance is:
C = As/V [Farad]
A = Ampere
s = seconds
V = voltage drop allowed
If we assume each pulse is eg. 1ms in the 500ms period, the current drain will be:
500ms/1ms*1µA = 500µA.
500µA in 1ms assuming an allowed voltage drop of 0.5V, will take a capacitor of:
0.0005 x 0.001 / 0.5 = 1µF
A tantalum capacitor will be best, since it has lower ESR than an electrolyte. You might need a bit higher value than calculated, as they have a large tolerance on the marked capacity.