How to speed up charging of output capacitors of ADXL335?
IF some interim DC value was superior you could divide C1 into two caps from Vcc and ground. While voltage on a cap cannot change instantaneously, two across the supply will approach instantaneous as anything there can. Equal caps give Vcc/2 at A0. Voltage is set to ~~= the divider set by the inverse of the capacitances. eg 10 nF to Vcc and 20 nF to ground gives (1/20 / (1/10 + 1/20) x Vcc = Vcc/3.

If you want the cap to more accurately approach the 1st out output you could add a FET switch across the 32 k resistor. This is turned on a power on and turned off "after a while. Depending on the voltage swing at Vout you MAY be able to use a single FET but for a fill range of Vout you will probably need a polarity agnostic switch - so, two FETS in series (both N or both P Channel), sources joined, gates joined, drains to either side of the 32 K resistor. (You need a very high value resistor gates-sources only if the gates are to be floated.   

Drive gates rail-rail with appropriate polarity for turn on/off for the FETs used. You MAY need a large value R from joined sources to ground to get switching going but quite possible not (based on my long ago playing). 

Or use a reed relay of similar - operated at power up, turn off "soon".

A transmission gate may suffice depending on equivalent channel resistance and tolerable current.