How did Kant know the correct time?

He knew the time because

the bells of a nearby church were ringing the hour

Kant set his clock at the usual time he left to see his friend just before he left.
Schmidt is a good friend so obviously his clock has the good time ( he probably saw Kant yesterday and adjusted his clock).
When he seee Kant today, he thinks that Kant still has his regular habits so Schmidt adjusts his clock.
The difference between the time before and the time after adjustment is also the difference between real time and Kant's clock so Kant can correct his clock.

Example :

Kant usually leaves at 8h so he sets his clock at 8h and leave. When he arrives it his 11h on Schmidt's clock and Schmidt adjusts it from 11h to 9h so Kant knows that his clock is 2 hours late.

Schmidt owns a watch/clock shop, and the purpose of Kant's visit was to pick up his repaired watch

Three possible answers:

1. He wound his clock before he left and walks the same route back. When he gets home he can then add two times the measured first journey time plus the stay time (assuming his friend's clock keeps good time and Kant walks at an extremely regular pace).
2. After staying a few hours from the evening he arrives home some time before dawn, he waits for sunrise, enters his house and sets his clock.
3. Similar to 2. He arrives after dawn and looks at his sundial.

I am guessing the expected answer is 1. due to the wording of the problem, specifically the uses of "no notion" and "prior".

To perform 1. he must measure his journey time from the two times his clock shows (when he leaves home and when he returns home) and the two times his friends clock shows (when he arrived there and when he left):

$Journey=(KC_\text{arrivedK}-KC_\text{leftK}-FC_\text{leftF}+FC_\text{arrivedF})/2$

$Time_\text{arrivedK}=2\times(Journey+FC_\text{leftF}-FC_\text{arrivedF})$

Possible answer:

The watch was repaired and delivered to his house in the hours that he was out. When he gets home, he simply sets the clock to his watch.

Possible answer:

As Kant arrives at his home next to the train station, he observes the daily train departing. With knowledge of the train schedule and the punctuality of the German trains, Kant confidently sets his clock to the correct time.

I think:

He burrowed or bought a watch from Schmidt (he is a merchant afterall).

This would be solvable if we have another source indicating the time. My solution-

Kant left the house as soon as he heard the church bell toll (e.g. - 10 am) (or a train leaving a station). As soon as he reaches, he checks the time (eg - 10:20 am). The difference tells him the time it takes from Kant to Schmidt's house (eg- 20 mins). He then leaves when the bell tolls again (eg- 4 pm), while noting the time. Since his gait is the same, it will take him the same time to return. He will then add this time to the time when he left Schmidt's house, to get the current time (4:20 pm).

I think the answer is simpler than everyone is making out. This following line of the question I think is the most important one:

"he walked with a slow, steady gait that had not varied in twenty years"

He would only need to count the number of steps from his friend's place to his home. Assuming he knew what this particular gait was, the time of making one pace, he would merely have to multiply the number of steps by this time. That was the time elapsed on his trip back home from Schmidt's house.

Now, the question states that he glanced at the clock upon arriving at Schmidt's house, he could either have added the time of his stay at Schmidt's to his arrival time to get his departure time, or otherwise made note of the time as he left (not specifically mentioned in the question).

Either that or you can bypass all the math by just assuming he passed a landmark close to his home and saw the time (which isn't stated explicitly), or his manservant had picked up the repaired watch (correctly set) and brought it to his home (neither stated explicitly).