How to select the set of last non-NULL values per column over a group?

Using the concatenation technique from The Last non NULL Puzzle by Itzik Ben Gan would look like this with your sample table and column data types.

select T.cat,
       cast(substring(
                     max(cast(T.t as binary(2)) + cast(T.val1 as binary(4))),
                     3,
                     4
                     ) as int),
       cast(substring(
                     max(cast(T.t as binary(2)) + cast(T.val2 as binary(1))),
                     3,
                     1
                     ) as char(1))
from #tab as T
group by T.cat;

Another way to write this query that divides the steps into CTE's to perhaps better show what is going on. It gives the exact same execution plan as the query above.

with C1 as
(
  -- Concatenate the ordering column with the value column
  select T.cat,
        cast(T.t as binary(2)) + cast(T.val1 as binary(4)) as val1,
        cast(T.t as binary(2)) + cast(T.val2 as binary(1)) as val2
  from #tab as T
),
C2 as
(
  -- Get the max concatenated value per group
  select C1.cat,
         max(C1.val1) as val1,
         max(C1.val2) as val2
  from C1
  group by C1.cat
)
-- Extract the value from the concatenated column
select C2.cat,
       cast(substring(C2.val1, 3, 4) as int) as val1,
       cast(substring(C2.val2, 3, 1) as char(1)) as val2
from C2;

This solution uses the fact that concatenating a null value with something results in a null value. SET CONCAT_NULL_YIELDS_NULL (Transact-SQL)

Just add a check for NULL in the partition will do

SELECT DISTINCT 
        cat, 
        FIRST_VALUE(val1) OVER(PARTITION BY cat ORDER BY CASE WHEN val1 is NULL then 0 else 1 END DESC, t desc) AS val1,
        FIRST_VALUE(val2) OVER(PARTITION BY cat ORDER BY CASE WHEN val2 is NULL then 0 else 1 END DESC, t desc) AS val2
FROM    #tab