With a voltmeter in series with the capacitor, it's going to charge kind of slowly through the 10-1000 megaohm impedance.
There isn't much practical difference between analog or digital meters for this job, since the signal of interest isn't rapidly varying. The current will start off high (limited by your series resistor, or the PSU's current limit), and then taper off to the current leaking thru the capacitor. As it reforms over the next several hours to days, the leakage normally decreases.