UPDATE only rows which match another table in SQL

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Yes, it is possible.

The reason some of your rows are getting NULLs updated is due to the subquery. When the subquery fails to find a matching row in tblB, the subquery returns NULL. But since the UPDATE has no WHERE clause, it will update all rows, even those where the subquery returns NULL. Consequently, to prevent this, you might do this:

UPDATE tblA A
SET A.Fld1 = ( SELECT B.Fld1
                 FROM tblB B
                WHERE A.Fld2 = B.Fld2 )
WHERE EXISTS ( SELECT B.Fld1
                 FROM tblB B
                WHERE A.Fld2 = B.Fld2 ) 

One potential issue may arise when the subquery actually finds more than one matching row in tblB. If this happens, the UPDATE will terminate with an error ("subquery may return only one row").

Another approach is to use a joined update query. The exact SQL that is required depends on which database system you're using, but here's an example for SQL Server:

update tblA
   set Fld1 = B.Fld1
  from tblA A
inner
  join tblB B
    on A.Fld2 = B.Fld2

Note that only rows that have a match are updated.