Is it possible to set starting count on binary counters like CD4020?
You could use a binary adder on the 4020 output lines. Then use the output of the adder instead of the direct output. You'd need something like a bunch of DIP switches to set the offset (that gets added). Then, when the 4020 is reset, the adder's output becomes the value set by the DIP switches (or MCU or whatever).
Here's a typical 4-bit adder in case it is useful to you: -

Picture from here and, that link also gives you more information on digital arithmetic that might help with your situation.
You should also note that with a carry-in and carry-out signal, these chips can be cascaded to 8 bits, 12 bits, 16 bits etc.: -

That picture is from here but originally came from above.