I guess you could pick a step-up regulator that has an ENable pin , which must be either connected to input voltage or to ground to turn on the regulator
then, figure out how your meter turns on (maybe there's two traces on the circuit board under that rotary wheel used to select ac voltage / dc voltage / resistance etc and you can just solder a wire to one of those traces.
OR, you could just program a 4-6 pin microcontroller (for example a 20-50 cent PIC10 or ATTiny) which would simply run 24/7 from battery using uA of energy and waiting for you to push a power on button ... when the button is pushed once, the micro can send voltage to the enable pin or connect the enable pin to ground and turn on the regulator.... then wait some time (for example 30minutes) and then turn off the regulator
or you could just use a modern step-up regulator with very low quiescent current, like 10-50uA, and just leave it running 24/7 - if the meter doesn't run, the discharge rate would be super low.
For example, there's MAX859 .. 0.8v .. 6v to maximum 6v output at up to 125mA :
https://www.digikey.com/product-detail/en/maxim-integrated/MAX859CSA/MAX859CSA-ND/1513203it has a 25uA quiescent current, or 1uA in shut down mode
The
MAX856 version that runs at higher frequencies and can do up to 0.5A output, stays at around 50uA quiescent current as long as the input voltage is reasonable (like 2.5v or higher, typical for 2 rechargeable NiMH batteries or alkaline batteries)
Another example would be
TPS61220 ... up to 6v output voltage with high efficiency and very low quiescent currents ..datasheet says typical 5uA