По сравнению с прошлым месяцем мой уровень русского резко повысился.

По сравнению с means in comparison with / compared with.

It is a fixed expression in Russian:

• по сравнению с прошлым месяцем = compared with last month

A learner should usually memorize по сравнению с as one chunk, because the case after it is predictable and the meaning is idiomatic.

Because after по сравнению с, the noun phrase goes in the instrumental case.

So:

• прошлый месяц = nominative
• с прошлым месяцем = instrumental

Forms here:

• прошлый → прошлым
• месяц → месяцем

That is why the full phrase is:

по сравнению с прошлым месяцем

Русского is in the genitive case because it depends on уровень.

Russian often uses the pattern:

• уровень чего? = the level of what?

So:

• уровень русского = level of Russian
• more fully, this is understood as уровень русского языка = level of the Russian language / my Russian level

This is very natural in Russian.

Yes. It is essentially a shorter version of:

• уровень русского языка

In everyday speech, Russians often omit языка when the meaning is obvious.

So these are similar:

• мой уровень русского повысился
• мой уровень русского языка повысился

The shorter version sounds natural and common.

Повысился is the past tense of повыситься, meaning to go up / to increase / to rise.

It has this form because it agrees with уровень, which is:

• singular
• masculine

So:

• masculine singular past = повысился
• feminine singular would be повысилась
• neuter singular would be повысилось
• plural would be повысились

Since уровень is masculine, we get:

мой уровень русского резко повысился

Because the sentence means that the level increased by itself, not that someone raised it directly as an object.

Compare:

• Я повысил уровень русского.
  = I raised/improved my Russian level.
  Here повысил is transitive: someone did the action to something.

• Мой уровень русского повысился.
  = My Russian level increased.
  Here повысился is intransitive/reflexive: the level itself went up.

In English, this is similar to the difference between:

• I raised it
• it rose / it increased

Both can work, but the nuance is a little different.

• повысился = went up / increased
• улучшился = improved / got better

With уровень (level), повысился is especially natural because levels are thought of as going up.

So:

• уровень повысился = very natural
• русский улучшился = also possible, but now the focus is more on your skill generally rather than your measured level

Yes, it is natural. Резко here means something like:

• sharply
• dramatically
• suddenly (depending on context)

So резко повысился means the level rose sharply or quite dramatically.

Other possible adverbs:

• значительно повысился = increased significantly
• сильно повысился = increased a lot
• заметно повысился = noticeably increased

Резко emphasizes a strong jump.

You can say things like мой русский улучшился, and that means my Russian improved.

But мой уровень русского is more explicit and sounds like you are talking about your proficiency level.

Compare:

• Мой русский улучшился. = My Russian got better.
• Мой уровень русского повысился. = My level of Russian increased.

The second sounds a bit more formal or measurable.

Because this opening phrase is a normal adverbial modifier, and in Russian it often does not need a comma.

So this is perfectly normal:

По сравнению с прошлым месяцем мой уровень русского резко повысился.

A comma might sometimes appear for intonation or stylistic reasons, but usually it is not required here.

For learners, the safest default is:

• no comma in this sentence

Yes. Russian word order is flexible, though different orders change emphasis.

Neutral version:

• По сравнению с прошлым месяцем мой уровень русского резко повысился.

Other possible orders:

• Мой уровень русского резко повысился по сравнению с прошлым месяцем.
• Мой уровень русского по сравнению с прошлым месяцем резко повысился.

All are understandable, but the original sentence is smooth and natural.

Starting with По сравнению с прошлым месяцем sets up the comparison first.

Russian past tense often covers both simple past and present perfect meanings from English.

So:

• повысился can mean increased
• but in context it may also be translated as has increased

Russian does not have a separate tense built exactly like the English present perfect. Context tells you whether the idea is:

• a past event, or
• a past change relevant now

Here the idea is very naturally:

• compared with last month, my Russian level has risen

No, but it is one of the most standard ways.

Some alternatives are:

• по сравнению с тем, что было в прошлом месяце...
• если сравнивать с прошлым месяцем...
• в сравнении с прошлым месяцем...

Still, по сравнению с прошлым месяцем is probably the most straightforward and idiomatic choice for a learner.

Yes, but it means something slightly different.

• По сравнению с прошлым месяцем... = compared with last month...
• За прошлый месяц... = over the last month / during the last month...

So:

• По сравнению с прошлым месяцем мой уровень русского резко повысился.
  The focus is on comparison with the previous state.

• За прошлый месяц мой уровень русского резко повысился.
  The focus is on the time period during which the improvement happened.

Both are natural, but they are not exactly the same idea.