The correct preposition to be inserted in the blank is:

(4) since

The completed sentence would be: "The school was not far from my home, and I used to walk, but since the start of last year, I had been going with the other girls in a rickshaw and coming home by bus."

In the context of the sentence, the word "since" is the appropriate preposition to indicate the starting point of an action or event. The phrase "since the start of last year" denotes a specific time when the speaker began going with the other girls in a rickshaw and coming home by bus. 

"Since" is commonly used to express a point in time from which an action or state has been continuous. Therefore, "since" is the correct preposition in this context.