Which of the following sentences is grammatically correct?

The correct answer is Option (3) → One should carry one’s own water bottle on a long train journey.

“One” is followed by “one’s”:

When a sentence begins with “one” in the sense of any person, the later pronoun must also be a form of “one”. So the sentence uses one’s (belonging to one). The sentence has no other error.

Why the Other Options are Incorrect?

Option 1: Dadi gave the old photo albums to my cousin and I. — Incorrect. The given sentence says “to my cousin and I”. This is wrong: “to” is a preposition, so the pronoun after it takes the object form “me”.
Corrected: Dadi gave the old photo albums to my cousin and me.

Option 2: Arjun hurt hisself while climbing the guava tree in the school garden. — Incorrect. The given sentence says “hisself”. There is no such form in standard English. The reflexive form for “he” is “himself”.
Corrected: Arjun hurt himself while climbing the guava tree in the school garden.

Option 4: The boy which sits next to me in class plays the tabla. — Incorrect. The given sentence uses “which” for “the boy”. “Which” is used for things and animals. For a person, “who” is needed.
Corrected: The boy who sits next to me in class plays the tabla.