A week after {it came toarrived at | I brought it into} my apartment, a fruit wasberry rippened.

“Fruit” isn’t wrong, but “berry” is more natural.

I was so happy when I saw it in the morning, but it had disappeared {in the evening when I got home | by the time I got home in the evening}.

Then, on thate weekend, I captured the suspecculprit with one of my cameras -- it was a turtledove.

I like the suspense of “then, on the weekend.” I changed “that” to “the” because it sounds smoother.

A week after it cameI brought it to my apartment, a fruit was ripppicked.

"Brought it" sounds better because you were the one who brought there, the plant didn't come by itself

Ripped is usually used when you talk about something being destroyed, however your plant's fruit was just taken. "Picked" is the usual verb when we talk about removing food from plants

I was so happy when I saw it in the morning, but it had disappeared in the evening when I got home.

Then on, that weekend, I captured the suspecculprit with one of my cameras-- it was a turtledove.

"Suspect" is who you think might have done a crime
If you are sure of the person, then this criminal is the "culprit."

A week after it came to my apartmentarrived, aI fruit was rippedound a ripe blueberry.

You can also say "a fruit was ripened", but it sounds a bit longer and more awkward.

Then on that weekend, I captured the suspect with one of my cameras-- it was a turtledove.

"Then on that weekend" is grammatically fine but sounds a little clunky - "that weekend" is more natural and concise.