Energy storage for AC inrush currents
There is no benefit to adding a capacitor across the AC supply (other than for power-factor correction - but that's another story).

You can control the switch-on using a zero-cross switch - generally a triac-based solid-state relay (SSR). While this will not reduce the inrush current of, say, a motor or cold filament lamp, it will prevent switch-on at peak voltage and eliminate EMF (electro-magnetic interference) and solves the problem of contact arc (by eliminating the arc).



Figure 1. An SSR circuit. Source All About Circuits.



Figure 2. Resultant waveform. Source: CCI Power.



Figure 3. A pre-packaged SSR.

The other option which may or may not be applicable depending on the load is to gradually increase the voltage using a non-zero-cross SSR.



Figure 4. By controlling the switch-on point in the mains half-cycle the RMS voltage can be adjusted from zero to almost 100%. Source: @Transistor.

In the case of Figure 4 the DC control signal would have to ramp up at a suitable rate.