Hortulana alvearium prope vineam ponit, ut apes ibi mel servent.

Because hortulana is in the nominative singular, which is the usual case for the subject in Latin.

The noun hortulana means a female gardener. The ending -a here matches a first-declension nominative singular form, so it tells you who is doing the action of ponit.

The form hortulana is the feminine version of the noun. A male gardener would be hortulanus.

So Latin is making the gardener explicitly female here:

• hortulanus = male gardener
• hortulana = female gardener

Alvearium is accusative singular, because it is the direct object of ponit.

The verb pono, ponere means to place, put, set, and the thing being placed goes in the accusative. So:

• hortulana = the subject, the one doing the placing
• alvearium = the thing being placed

Because prope is a preposition that takes the accusative case.

So:

• prope vineam = near the vineyard

Even though English uses near without changing the noun, Latin requires the accusative after prope. That is why vinea becomes vineam.

It is accusative singular.

It is accusative because it follows prope, and prope takes the accusative. So vineam is not the direct object of the main verb; it is the object of the preposition prope.

Because Latin often expresses purpose with ut + a subjunctive verb, where English often uses to or in order to.

So:

• English: She places the hive near the vineyard to store honey
• Latin: ... ponit, ut apes ibi mel servent

More literally, the Latin means so that the bees may store honey there.

This is called a purpose clause.

Because it is in a purpose clause introduced by ut.

A very common Latin pattern is:

• ut
  • subjunctive = so that, in order that

So servent is not just saying that the bees do store honey as a simple fact; it expresses the gardener's purpose in placing the hive there.

Because servent is the present subjunctive, while servant is the present indicative.

From servo, servare:

• servant = they store / they keep
• servent = they may store / they should store

In a purpose clause after ut, Latin uses the subjunctive, so servent is the correct form.

Apes is nominative plural, because it is the subject of servent.

Inside the ut clause, the bees are the ones doing the storing, so apes must be the subject of that clause.

This is a good reminder that a sentence can contain more than one subject:

• hortulana is the subject of ponit
• apes is the subject of servent

Mel is accusative singular here, because it is the direct object of servent.

The bees are storing honey, so mel is the thing being stored.

A small detail: mel is a neuter noun, and its nominative and accusative singular forms look the same. So the form itself does not change, but its job in the sentence is accusative.

Ibi is an adverb meaning there.

It tells you where the bees store the honey:

• apes ibi mel servent = so that the bees may store honey there

Because it is an adverb, it does not have a case.

We know from the combination of ut with a subjunctive verb and from the sense of the sentence.

Here, the gardener places the hive near the vineyard for a purpose. That makes ut a marker of purpose:

• ut
  • subjunctive = so that, in order that

So ut is not introducing a result here, and it is not being used in some other sense such as as or when.

Latin word order is more flexible than English word order. Verbs often come at or near the end of their clause, although they do not have to.

So:

• Hortulana alvearium prope vineam ponit

is a very natural Latin order. The endings show what each word is doing, so Latin does not depend on word order as heavily as English does.

Yes. This is the normal sequence for a present main verb followed by a purpose clause.

• ponit is present indicative
• servent is present subjunctive

After a primary tense such as the present, Latin normally uses the present subjunctive in a purpose clause. So ponit ... ut ... servent is exactly what you would expect.