Is it grammatically correct to omit prepositions sometimes?

Yes, you can omit prepositions in these two cases.

In this case, we are omitting a preposition, but more importantly turning these verbs from the intransitive forms into their transitive ones. Therefore, I wouldn't refer to it as "omitting a preposition," but rather "using the transitive form."

Transitive verbs take a direct object, whereas intransitive verbs do not (but they can still take indirect objects).

An example of a transitive verb is "gave" in the sentence "I gave the book to him." "Book" is the direct object it is directly affected by the action of the verb. "Him" is the indirect object because it is indirectly affected by the action of the verb.

An example of an intransitive verb is "spoke" in the sentence "I spoke to him yesterday." "Him" is the indirect object because of the preposition "to."

"Graduate" and "fly" have both transitive and intransitive forms. When we remove the prepositions in the sentences from the question, we will use the transitive form instead of the intransitive one. If the two verbs did not have transitive forms, we would not be able to remove the preposition (e.g. "I spoke to him" cannot become "I spoke him").

"Graduate" has many definitions, but the applicable ones are:

[intransitive] Successfully complete an academic degree, course of training, or (in North America) high school. [Lexico]

and

[transitive] Receive an academic degree from. [Lexico]

I found a definition for intransitive "fly," but I couldn't find one for transitive "fly."

However, I found examples all over the internet (e.g. "fly JetBlue" or "fly first class"), and I even found one on the ELU site.

In fact, the transitive version of "fly" is actually more popular than its intransitive one (in this context). Even though I couldn't find an actual definition for transitive "fly," it is still an established word.

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If you want to read a little more about this topic, read this article.