Этот график мне подходит.

Because подходить in this meaning takes the person in the dative case.

So Russian says something more like:

• This schedule suits to me
• This schedule works for me

That is why you get:

• мне = to me / for me

Compare:

• Я = I subject form
• Меня = me accusative/genitive
• Мне = to me / for me dative

With verbs like подходить, нравиться, мешать, and some others, English speakers often have to get used to Russian using the dative where English uses I/me in a different way.

Here подходит means suits, is suitable, or works.

In this sentence, it does not mean physical movement like is approaching. Russian подходить can have several meanings depending on context:

• Поезд подходит к станции. = The train is approaching the station.
• Этот цвет мне подходит. = This color suits me.
• Этот график мне подходит. = This schedule works for me.

So the idea here is suitability, not movement.

Because график is a masculine singular noun, and этот must agree with it in gender, number, and case.

So:

• этот = masculine singular
• эта = feminine singular
• это = neuter singular
• эти = plural

Examples:

• этот график = this schedule
• эта книга = this book
• это письмо = this letter
• эти документы = these documents

Since график is masculine, этот is the correct form.

It can mean either, depending on context.

Common meanings of график include:

• schedule / timetable / work schedule
• graph / chart

In the sentence Этот график мне подходит, the most natural meaning is often This schedule works for me.

But in another context, it could also mean something like This graph is suitable for me or This chart works for my needs, though that is less common in everyday conversation.

So learners should remember that график is a context-dependent word.

Подходит is:

• 3rd person singular
• present tense
• imperfective aspect

Why 3rd person singular? Because the subject is этот график = this schedule, which is singular.

So the verb agrees with график, not with мне.

Basic pattern:

• Этот график подходит. = This schedule suits / works.
• Эти графики подходят. = These schedules suit / work.

Notice the plural change:

• singular: подходит
• plural: подходят

Because мне is not the grammatical subject.

The subject is этот график. That is the thing doing the suiting.

So Russian structure is roughly:

• This schedule suits me
• subject: этот график
• indirect object: мне

That is why the verb is singular:

• Этот график мне подходит.

If the subject were plural, the verb would be plural:

• Эти графики мне подходят.

Yes. Russian word order is flexible, though the neutral order here is:

• Этот график мне подходит.

You could also say:

• Мне подходит этот график.
• Этот график подходит мне.

The meaning stays basically the same, but the emphasis changes a little.

For example:

• Мне подходит этот график. = emphasis on this schedule
• Этот график мне подходит. = neutral, natural statement
• Этот график подходит мне. = possible, but often sounds a bit more marked or contrastive

For learners, the original order is a very good standard pattern.

Not in this sentence.

That is a very common learner question, because подходить can literally mean to approach / come up to in other contexts. But Russian often uses the same verb in multiple related meanings.

Here the meaning is idiomatic:

• мне подходит = suits me / works for me

A useful shortcut is:

• if подходить is used with a person in the dative and a thing as the subject, it often means to suit / to be suitable

Examples:

• Мне подходит этот размер. = This size suits me.
• Тебе подходит это время? = Does this time work for you?
• Нам подходит такой вариант. = That option works for us.

Just add не before the verb:

• Этот график мне не подходит. = This schedule doesn’t work for me.

This is the normal way to negate it.

Other examples:

• Этот цвет мне не подходит. = This color doesn’t suit me.
• Такое время мне не подходит. = That time doesn’t work for me.

Yes, there is some overlap, but they are not always identical.

• мне подходит = it suits me / it works for me / it is suitable for me
• меня устраивает = it is acceptable to me / I am satisfied with it

Very roughly:

• подходит focuses more on fit, suitability, convenience
• устраивает focuses more on being acceptable or satisfactory

Examples:

• Этот график мне подходит. = the schedule fits my needs/availability
• Этот график меня устраивает. = I am okay with this schedule; it is acceptable to me

In many real situations, both are possible, but подходит is especially natural when talking about schedules, times, sizes, colors, and practical compatibility.

The stress is:

• Э́тот гра́фик мне подхо́дит.

A rough pronunciation guide:

• Э́тот = EH-tuht
• гра́фик = GRA-fi k
• мне = mnyeh
• подхо́дит = pa(d)-KHO-deet

The most important stress points are:

• Э́тот
• гра́фик
• подхо́дит

If you ignore vowel reduction and just place the stress correctly, you will already sound much better.

Yes, very often. This is a highly useful pattern in Russian:

• [thing] + мне/тебе/ему... + подходит

Examples:

• Это время мне подходит. = This time works for me.
• Этот размер тебе подходит? = Does this size suit you?
• Новая работа ему подходит. = The new job suits him.
• Нам подходит этот вариант. = This option works for us.

This is a great sentence model to memorize.