Я часто думаю о нём вечером.

Because думать in the meaning to think about normally takes о + Prepositional case:

думать о ком? о чём? = to think about whom? about what?

So:

• о нём = about him
• о ней = about her
• о тебе = about you

Using его here would be wrong, because его is not the form required after о in this meaning.

нём is the Prepositional singular form of он.

The full pattern is:

• он
• о нём

Russian third-person pronouns often add an initial н- after prepositions:

• у него
• к ней
• с ним
• о нём

So the н- is not random; it is a regular feature of these pronoun forms after prepositions.

Yes. The correct pronunciation here is нём, with stress on ём.

In textbooks and learner materials, it is usually written as нём to make that clear. In ordinary Russian writing, people often omit the dots over ё and write нем, but it is still usually understood and pronounced as нём from context.

So:

• careful/learner spelling: о нём
• common everyday spelling: о нем

Here it means I think about.

That is because of the structure думать о ком-то / о чём-то = to think about someone / something.

Also, думаю is:

• 1st person singular
• present tense
• from the imperfective verb думать

In this sentence, because of часто, it has a habitual meaning: I often think about him.

Without о + Prepositional, думать can also mean to think / believe / suppose, depending on context.

вечером is the Instrumental singular of вечер.

Russian often uses the instrumental case without a preposition for parts of the day when talking about time:

• утром = in the morning
• днём = in the daytime / during the day
• вечером = in the evening
• ночью = at night

So вечером here simply means in the evening. This is a very common time expression pattern.

Yes, often you can.

Russian verbs usually show the person clearly, and думаю already means I think. So Часто думаю о нём вечером is possible.

However, я is still very commonly used. Including it can make the sentence:

• clearer
• more complete
• slightly more explicit
• more contrastive if stressed

So both are possible:

• Я часто думаю о нём вечером.
• Часто думаю о нём вечером.

No. Russian word order is fairly flexible.

The sentence Я часто думаю о нём вечером is a natural, neutral order. But other orders are also possible, with different emphasis:

• Вечером я часто думаю о нём.
  Focuses more on in the evening.

• О нём я часто думаю вечером.
  Brings about him to the front for emphasis.

• Я о нём часто думаю вечером.
  Also possible, with a slightly different rhythm or focus.

What stays fixed is that the preposition must remain with the pronoun: о нём.

Yes, it could move, but Я часто думаю... is the most neutral and natural placement.

In Russian, adverbs like часто are flexible, but different positions can sound more or less natural depending on emphasis.

For example:

• Я часто думаю о нём вечером.
  Neutral.

• Я думаю о нём часто.
  Possible, but it emphasizes often more strongly.

• Часто я думаю о нём вечером.
  Also possible, but more marked.

So the given position is simply the most standard one.

Both can mean about him, but they are not always equally natural.

With думать, the most standard and neutral choice is о нём:

• думать о нём

You may also hear:

• думать про него

but про него is generally more colloquial. It is very common with verbs like говорить or рассказывать, but with думать, о нём is usually the best choice for learners.

By itself, вечером means in the evening as a time frame.

Because the sentence also contains часто, the overall meaning is usually habitual: the speaker often thinks about him during the evening time.

It does not necessarily mean every evening. If you want that idea clearly, Russian would more likely say:

• каждый вечер = every evening

So:

• Я часто думаю о нём вечером = a general repeated habit
• Я думаю о нём каждый вечер = every evening

Yes, and it changes the nuance a little.

• вечером = in the evening
• по вечерам = in the evenings / on evenings / evenings in general

Since your sentence already has часто, вечером works well. But по вечерам can sound even more clearly habitual:

• Я часто думаю о нём по вечерам.

That sounds like I often think about him in the evenings as a repeated pattern.