How do I specify the amount of syrup taken from this bottle?

You can refer to a single "amount" as a pump.

Each 7.5 oz. Dial® Liquid Hand Soap bottle provides about 125 pumps per bottle.

Link

Though its OED definition is a bit vague and not as strictly defined, you can find a justification for this usage of "pump" as "the amount produced by a single pumping action":

1.1 (in singular) An instance of moving something by or as if by a pump.
‘the pump of blood to her heart’

Link

Though this speaks about the general act of pumping, rather than a countable noun.

However, it seems within the range of acceptability that the produced amount can be referred to by the act of pumping itself.

• One pump = the produced amount from pumping once.
• Two pumps = the produced amount from pumping twice.

It's shorthand for counting how often the act of pumping occurs. Similar usages can be found:

• Two slices of cheese (you slice twice)
• A pinch of salt (you pinch once)

However, for these examples, their nouns have been well-defined in the OED by now. I believe that the analogous "pump" simply has not been in common usage enough to warrant adding its definition to the OED.

But its correctness should be analogous to that of "slice" or "pinch". From the comment that was posted, "squeeze" and "squirt" are equally correct for the same reason: it uses the act of doing something to describe the produced result from this act.

Coffee shops in the UK tend to refer to a shot of syrup.

This presumably originated with the term for a measure of spirits, but is also used for a serving of espresso. That's a similar volume to a serving of syrup so the it's not surprising that the use propagated.