In Postgres 9.1:
SELECT name, (f).*  -- note the parentheses!
FROM  (SELECT name, calculate_payments(id) AS f FROM person) sub;
Assuming that your function has a well-defined return type with column names (id, action, amount). And that your function always returns the same id it is fed (which is redundant and might be optimized).
The same in much more verbose form:
SELECT sub.id, sub.name, (sub.f).action, (sub.f).amount  -- parentheses!
FROM  (
   SELECT p.id, p.name, calculate_payments(p.id) AS f(id, action, amount)
   FROM   person p
   ) sub;
Set-returning functions in the SELECT list result in multiple rows. But that's a non-standard and somewhat quirky feature. The new LATERAL feature in pg 9.3+ is preferable.
You could decompose the row type in the same step:
SELECT *, (calculate_payments(p.id)).*  -- parentheses!
FROM   person p
But due to a weakness in the Postgres query planner, this would evaluate the function once per result column:
Or in your case:
SELECT p.id, p.name
     , (calculate_payments(p.id)).action
     , (calculate_payments(p.id)).amount
FROM   person p
Same problem: repeated evaluation.
To be precise, the equivalent of the solution in pg 9.3+ is this, preserving rows in the result where the function returns 0 rows:
SELECT p.id, p.name, f.action, f.amount
FROM   person p
LEFT   JOIN LATERAL calculate_payments(p.id) f ON true;
If you don't care about this, you can simplify in pg 9.3+:
SELECT p.id, p.name, f.action, f.amount
FROM   person p, calculate_payments(p.id) f;
Closely related: