As is typical for a logic puzzle, there are many possible paths to solving the grid. One possibility is given below:
We are told that Olive starts yodeling practice at 2:00, Roger starts at the factory at 3:00, and a Grant starts at the idyll at 4:00
It is given that Lewis Heffley ends at 4:00. Since each task lasts at least one hour, Lewis must start by 3:00. Since Olive and Roger start at 2:00 and 3:00, Lewis Heffley must work from 1:00-4:00.
The task at the garden will be finished last, and the last task finishes at 9:00. To finish at 9:00, it must begin at either 4:00 or 5:00. Since the 4:00 task is at the idyll, the 5:00 task must go until 9:00 at the garden. We are also told that this last task is listing duties.
Since the first task finished will be yodeling practice, which begins at 2:00, and the 1:00 task ends at 4:00, yodeling practice must be only one hour and end at 3:00. We know that the first task to end must be at the rectory, so the 1:00 task must be in the only remaining location, the showroom.
The two-hour and five-hour tasks remain. Since the 4:00 task cannot end at the same time as the 5:00 task, it must end at 6:00, and the 3:00 task must go until 8:00.
Olive cannot be Bardin or Forrester. The only remaining surname is Thompson. We know that one of Bardin or Forrester must work in a location that starts with that letter. No locations begin with B, so Forrester must be in the factory at 3:00 and Bardin in the Garden at 5:00.
Similarly, one of Coral and Idaho must work in a location which shares a starting letter with their name, so Idaho must be in the idyll at 4:00 and Coral in the garden at 5:00.
The third task finished is cutting tape, which must start at 4:00. Sawing logs cannot happen in the showroom, so it must happen in the factory, leaving planting turnips to happen in the showroom.
This give the following completed grid:
The trick to extracting the answer involves noting the elapsed time of each task (not given a column in the grid). These times can be used to extract a given letter from each answer by taking the letter in the position equal to the length in hours of the task.
Reading down the columns, these letters give the phrase WORD AFTER DAY, NUT, OR ODD. The word which can follow each of these three to make a sensible phrase is JOB.