Fill in the blank with the appropriate word :You have played a great…

The correct answer is Option (3) → without

"Without" meaning "if it had not been for":

• Without is a preposition. Followed by a noun phrase ("your help"), it can state an unreal condition: without your help = if you had not helped me.
• Deciding clue: the main clause "I possibly would have landed myself in a problem" is the imagined result of that condition.
• This sentence: the speaker thanks the listener ("You have played a great role") because the help saved him from trouble.
• Complete sentence: You have played a great role, for without your help I possibly would have landed myself in a problem.
  Example: Without your advice, I would have made a big mistake.

Why the Other Options are Incorrect?

Option 1: unless — Incorrect. Unless is a conjunction meaning "if not"; it needs a full clause with a verb (unless you had helped me), not only a noun phrase.
Example: You will miss the bus unless you hurry.

Option 2: after — Incorrect. After shows time; "after your help I would have landed in a problem" would make the help the cause of the trouble.
Example: We went for a walk after lunch.

Option 4: despite — Incorrect. Despite means in spite of; it would say that the help failed, which goes against "You have played a great role".
Example: Despite the rain, the match went on.

Contrast: "without", a preposition, takes a noun phrase ("without your help"); "unless", a conjunction, needs a full clause ("unless you had helped me").