Learn Russian: Утром у меня есть время.

Russian doesn't use a direct verb for "to have". Instead, it uses a phrase that literally translates to "At me there is". У means "at" or "by", меня is "me", and есть means "there is".

Why do we use меня here instead of я?

The preposition у always requires the word following it to be in the genitive case. Меня is the genitive form of the pronoun я (I).

The dictionary word for "morning" is утро. Why does it end in -ом here?

To say "in the morning", Russian doesn't use a preposition like "in". Instead, it changes the noun's ending. Adding -ом turns the noun утро into the time expression утром (in the morning). You will see this pattern with other times of day too, like вечером (in the evening).

Since "time" is what I have, shouldn't время change its ending like an object?

No, because in Russian you are literally saying "At me is time." This means время (time) is actually the subject of the sentence doing the "being", not the object receiving an action! Because it is the subject, it stays in its basic dictionary form.