Here is idiomatically correct version of your code that works:

f[x_] := x^2 /; a > 0

Block[{a = 1}, f[2]]

The approach you tried doesn't work because f[2] is evaluated before the replacement rule is applied.

An approach like the following certainly works:

Module[{fResult},
 a = 1;
 fResult = f[2];
 a =.;
 fResult
 ]

But that's probably too ornate for whatever you are doing given that you don't want to define an f[x_,a_].

Another approach is to define f as

f[x_] := x^2

and then to execute

If[a > 0, f[2]] /. a -> 1

or

If[a > 0, f[2],alternative] /. a -> 1

I'm sure there are lots of approaches, but which one you might use depends on the particular problem that you are coding.

Why so complicated? How about simply:

f[x_] := If[a > 0, x^2, "whatever"]

this works exactly as intended:

a=.;
f[2] /. a->1

Yours Henrik