In my solution I assumed your data column is A1:A5, and the amounts are
in B1:B5. The dates are a question mark, tho: are they custom
formatted "Date" type data fields, or "Text" type data fields?

If they are dates, the formula is:
=SUMPRODUCT(--(MONTH(A1:A5)=1),B1:B5)
This is for January, month 1. You can create results for other months
by changing the month number in the formula.

If the dates are text, the formula is:
=SUMPRODUCT(--(MID(A1:A5,1,2)="01"),B1:B5)
This formula performs some text parsing; if you need to get results for
other months replace "01" with two digit months numbers.