Your solution using total unchanged method is correct.

Alternative method:

We know at the beginning Calvin has 4 units. 

Then we know at the end, what’s left for each person is as follows:

Calvin       [3 units of Calvin][1 part of Susie]         =   45

Susie         [4 parts of Susie]  = 36

If 4 parts of Susie = 36, 1 part of Susie = 9.

3 units of Calvin = 45 – 9 = 36 

Hence 4 units of Calvin at the beginning = 48

Yes! Your answer is correct.

The working can perhaps be simplified. Just focus on the movement of +/- 9 and +/- 12 cards. Seems more natural for the problem. (Just my opinion.)

The total sum of 81 is not quite necessary in the working – but no harm tracking it as it is useful for sanity check.

After Susie gave  1/5 of her cards to Calvin, she left with 36 cards.

So Susie left with  4/5 of her cards after giving.

Susie has    5/4 * 36 = 45 cards before giving.

Calvin has 45 cards after Susie gave   1/5* 45 = 9 cards to him.

So he has 45-9=36 cards before giving  1/4 of his cards to Susie.

Or Calvin has   4/3* 36 = 48 cards at first.

Checking:

Susie has 45 cards after receiving   1/4 * 48 = 12 cards  Or Susie has 45-12=33 cards at first.

                                                                                                Calvin                          Susie

First                                                                                         48                                33

After Calvin gave   of his cards to Susie               3/4    * 48 = 36            33 + (1/4  *48 )=45  

After Susie gave   of his cards to Calvin            36 + (1/5   * 45) =45                  4/5  * 45 = 36