kondo zikan ga dekitara, tosyokan de issyo ni nihongo wo benkyousimasen ka?

In this sentence, 今度 means something like next time, sometime soon, or when the opportunity comes up.

It does not have to mean one exact future occasion. It often has a slightly vague, natural feel, like:

• when you have time next
• sometime soon
• next chance you get

So 今度 時間ができたら gives the idea of if/when you get some free time sometime soon.

Yes, できる often means can do, but it also has another very common meaning: to come into existence, to be made, or to become available.

So 時間ができる means:

• time becomes available
• you get some free time
• you end up having time

That is why 時間ができたら naturally means if/when you have time.

It is more natural in Japanese than trying to say something like if you make time in many everyday situations.

Because 時間 is the thing that becomes available.

With できる, the thing that comes into existence or becomes possible often takes が:

• 時間ができる = time becomes available
• 予定ができる = plans get made
• 仕事ができる = work comes up / can do work, depending on context

So here, 時間が is not the direct object. It is the thing that appears or becomes available.

～たら is a conditional form. In this sentence, it means if or when.

So:

• 時間ができたら = if you get some free time
• or more naturally, when you have time

In real conversation, ～たら often covers both ideas. English forces you to choose between if and when, but Japanese is often more flexible.

Here it sounds natural and friendly because the speaker is talking about a possible future situation.

Because で marks the place where an action happens.

• 図書館で勉強する = study at the library

If you used に, it would usually suggest a destination or location of existence, not the place where the studying happens.

Compare:

• 図書館に行く = go to the library
• 図書館で勉強する = study at the library

So で is correct because 勉強する is the action taking place there.

一緒に means together.

It shows that the speaker is inviting the other person to do the activity with them:

• 一緒に日本語を勉強しませんか = Won’t you study Japanese together with me/us?

Without 一緒に, the sentence would still work, but it would lose some of the clear invitation feeling. 一緒に makes it obvious that this is a shared activity.

Because 日本語 is the thing being studied, so it is treated as the direct object of 勉強する.

• 日本語を勉強する = study Japanese

This is the standard pattern.

You may also see:

• 日本語の勉強をする = do Japanese study / study Japanese

Both are correct, but 日本語を勉強する is very common and straightforward.

Yes, it is literally negative:

• 勉強しませんか = won’t you study?

But in Japanese, this kind of negative question is a very common and polite way to make an invitation or suggestion.

So it really means something like:

• Would you like to study Japanese together?
• How about studying Japanese together?

It sounds softer and less pushy than a direct command or a very blunt suggestion.

This is a very useful pattern:

• 行きませんか = Would you like to go?
• 食べませんか = Would you like to eat?
• 話しませんか = Would you like to talk?

They are quite different.

• 勉強しますか usually means Will you study? or Do you study? depending on context.
• 勉強しませんか usually means Would you like to study? as an invitation.

So:

• 勉強しますか asks about the other person’s action
• 勉強しませんか invites them to do it together

That is why しませんか is the natural choice here.

Japanese often leaves out subjects when they are clear from context.

In this sentence:

• 時間ができたら usually implies if you have time or if you get some free time
• 勉強しませんか implies shall we study? or would you like to study with me?

So English needs words like you and we, but Japanese often does not.

That omission is completely normal. In fact, adding the pronouns can sometimes sound less natural unless there is a reason to emphasize them.

It sounds polite, friendly, and natural.

Why?

• 勉強しませんか is a polite invitation
• the sentence is not too stiff
• it sounds suitable for classmates, friends, coworkers, or someone you want to be polite to

It is softer than a direct suggestion like:

• 一緒に日本語を勉強しよう = Let’s study Japanese together

That version is more casual. The original sentence is a good polite everyday invitation.

A natural reading is:

こんど じかん が できたら、としょかん で いっしょ に にほんご を べんきょうしません か？

A few pronunciation notes:

• 今度 = こんど
• 図書館 = としょかん
• 一緒 = いっしょ
• 勉強 = べんきょう

So the whole sentence is:

こんど じかん が できたら、としょかん で いっしょ に にほんご を べんきょうしません か？