Whilst you certainly can use MySQL's IF() control flow function as demonstrated by dbemerlin's answer, I suspect it might be a little clearer to the reader (i.e. yourself, and any future developers who might pick up your code in the future) to use a CASE expression instead:
UPDATE Table
SET A = CASE
WHEN A > 0 AND A < 1 THEN 1
WHEN A > 1 AND A < 2 THEN 2
ELSE A
END
WHERE A IS NOT NULL
Of course, in this specific example it's a little wasteful to set A to itself in the ELSE clause—better entirely to filter such conditions from the UPDATE, via the WHERE clause:
UPDATE Table
SET A = CASE
WHEN A > 0 AND A < 1 THEN 1
WHEN A > 1 AND A < 2 THEN 2
END
WHERE (A > 0 AND A < 1) OR (A > 1 AND A < 2)
(The inequalities entail A IS NOT NULL).
Or, if you want the intervals to be closed rather than open (note that this would set values of 0 to 1—if that is undesirable, one could explicitly filter such cases in the WHERE clause, or else add a higher precedence WHEN condition):
UPDATE Table
SET A = CASE
WHEN A BETWEEN 0 AND 1 THEN 1
WHEN A BETWEEN 1 AND 2 THEN 2
END
WHERE A BETWEEN 0 AND 2
Though, as dbmerlin also pointed out, for this specific situation you could consider using CEIL() instead:
UPDATE Table SET A = CEIL(A) WHERE A BETWEEN 0 AND 2