Is there a shortened way to say specific numbers in Japanese?

for example, 78 would be read as nanajuu-hachi but is it correct slang to shorten it to nana-hachi? Like in English 4,500 is read as four-thousand five-hundred but can be read as forty-five hundred in slang.

No, I don't think there's any such thing.

The only alternative that I know of is to read a number as a string of digits, e.g. 4500 as yon go maru maru, or 123.45 as ichi ni san ten yon go, which you might do when dictating a readout or result to someone, and also, apartment numbers and phone numbers are normally read as strings of digits.