Yes, they are all correct. The only issue, as Fabi points out, concerns using “was” or “is” in the last sentence. “Was” is correct if, as of the time the speaker is speaking, the father should already have purchased the tickets. “Is” is correct if the father is supposed to purchase the tickets sometime before going to the train station, even if the purchase is to occur after the speaker is speaking.

He "is" supposed to buy train tickets in advance.

They are good stand alone sentences. Comments We rarely use “a.m.” or “p.m.” in speech. “Saturday at 7” or “Saturday morning at 7” if not clear from context. He was/is supposed to buy his ticket(s) in advance. Also, the sentences are good by themselves, but they don’t fit together smoothly, if that’s the intention. What are you doing this Saturday? In the morning I’m going fishing on the river, but I have to clean my house in the afternoon. Later, at 8, I have to take my father to the train station. Which reminds me, he’s supposed to buy his ticket in advance. I better send him a text so he doesn’t forget.

I think these are all very good the way they are. It may not be perfect formal wording, but it is good everyday English.

Sorry I inputted my comment in the wrong spot.