Can I install a sprinkler system timer underground?
You would not use the existing good sprinkler wires to bring the 120VAC out to the valve box. The wires are not rated and approved for that type of use. Place the converter transformer indoors to step the 120VAC down to the 24VAC. The "good" wires could then be used to route the 24VAC outdoors.

I would not use the sprinkler timer in the valve box. It would be very inconvenient to use it in that manner and the underground environment is highly likely to not be friendly to the control unit circuitry. Instead you should install an outdoor timer unit on an exterior wall above the valve box. (Use "outdoor sprinkler timer" in Google to see a selection of possible timers). The better outdoor timers have a lid that closes over the controls to protect the interior from the elements. You could then use a PVC conduit to route the necessary wires from the timer down into the valve box to keep things neat.

Do select the timer as one that works off the low voltage.