Presumably given mention of INDIRECT & SUMIF the Budget file is open at all times ?
(If not both of the above are precluded)

You can use INDEX/MATCH or VLOOKUP, in both cases using a MATCH to determine appropriate column.

Ignoring the external references for sake of simplicity - assume your table above is on a sheet called BUDGET in A1:M100 with months in B1:M1 and codes in A2:A100 - values in resulting matrix...

=INDEX(BUDGET!$A$2:$M$100,MATCH($A10,BUDGET!$A$2:$A$100,0),MATCH($G$9,BUDGET!$B$1:$M$1,0))
where A10 holds unique code of interest

In VLOOKUP terms

=VLOOKUP($A10,BUDGET!$A$2:$M$100,MATCH($G$9,BUDGET!$A$1:$M$1,0),0)

I prefer INDEX/MATCH myself as it's more flexible (and can also result in fewer precedent cells / calculations).