1. He is helping an old man get on the bus.

2. He is helping an old man get onto the bus.

(Which preposition do I have to use? are both OK?)

3. Look at the picture. He is trembling because of cold. It is winter. It is windy as well. Look at the bare (naked, leafless) tree. What advice can you give the person who is in the picture. He is wearing a shirt. So you can give advice to him like this.
Wear a coat, and you'll feel warmer.

(Is the passage grammatical? Would you correct any errors?)

4. He stayed at the Green Park Hotel.

5. He stayed at Green Park Hotel.

(Which one is correct? Do we have to use 'the' or not?)

1 and 2. Both prepositions are correct, but "on" is more common in this sentence.

3. It's good, but you need a question mark after picture. ". . . person who is in the picture?"

4. is better. We usually use "the" before the names of hotels.