Ways to measure current in picoamperes
Power the micro-controller with a capacitor, charged to a known voltage. Wait an appropriate amount of time, then measure the voltage. Calculate the current from the delta-V and the C. (Don't measure the voltage continuously, unless you have a meter with a high-enough impedance, because that might draw extra current.) You will need a capacitor with known capacitance, but in a pinch you could measure a capcitor in the same way by discharging it through a known resistor.

As the comments point out, other current paths might contribute to the discharge of the capacitor (including self-discharge). You could repeat the measurement with the UC removed and see what value that gives. Then you might think about whether you can realisticly avoid such 'other' currents in your design.

And don't forget your batteries self-discharge and/or ageing!

If you aim is too 'see' the power-down mode of the chip in action you could use the capacitor, build a simple circuit that periodically connects it to the power supply (if possible synchronysed with the uC's activity cycle, must have a realy low leakage current!), and watch the C's voltage on a scope (the scope impedance must be higher than the UC's current consumption, or you might even use AC coupling if the uC"s activity cycle is short enough). This way you can verify both the time-wise division in high and low current consumption, and the currents in both modes.