I've got several sentences that include 'of the past' or 'in the past' written in my book
Phones of the past were all landlines.
The main difference from transportation in the past.
Many restaurant in the past were quite run down in terms of appearance.
I have no idea what is the difference. Could you help me get out of this tight spot?
Of the past is an adjectival phrase, but in the past is an adverbial phrase. So your first sentence, if you substituted one phrase for another, would have different meanings. To say that "phones in the past were landlines", which contains the omitted verb "used" before "in the past" does not suggest that modern phones are landlines, but "phones of the past were landlines" says that those which have survived in someone's basement still are landlines.
This is implicit in your second sentence fragment (which is not a complete sentence), since presumably the rest of the sentence compares transportation in the past to transportation today, and either discusses the differences between them or establishes similarities - but either way the two classes of transportation are assumed to be different. If they were not different there would be no point in comparing them. But "transportation of the past", such as horse-drawn carriages or steam-driven locomotives, has not changed today, although it may not be used much.
I don't believe there's really any substantive difference between the two in the sentences you posted - in those examples, you could quite easily swap each of the bolded phrases and the audience would still understand what was meant.
In a technical sense, however, 'Of' refers to 'being from'. I am of Australia. However, I am not necessarily always in Australia. I might be elsewhere. In your examples, though, they're fixing the point of reference as 'the past', so the subjects are of, and in, the past. Hence, either one will work.
1) Phones of the past were all landlines.
of the past is OK because it is modifying things.
2) The main difference from transportation in the past...
3) Many restaurants in the past were quite run down in terms of appearance.
These are not things, so I would use in the past. Restaurants are establishments with a building, equipment, supplies, customers, etc. Transportation is a system of moving people or things from one place to another.
