ביקשתי ממנה לבוא בשבע ורבע, כדי שנוכל לשתות קפה לפני הפגישה.

ביקשתי is the past tense, first person singular form of לבקש (to ask / to request).

So:

• ביקשתי = I asked / I requested

The ending ־תי is a very common past-tense ending for I in Hebrew. Because the verb already shows the subject, Hebrew usually does not need the pronoun אני.
So ביקשתי by itself already means I asked.

Yes — ממנה literally means from her, but with the verb לבקש, Hebrew often uses the pattern:

לבקש מ־מישהו לעשות משהו
= to ask someone to do something

So:

• ביקשתי ממנה = I asked her
• literally, more like I requested from her

This is just the normal Hebrew structure. English says ask her, but Hebrew commonly says ask from her.

ממנה is the preposition מ־ / מן (from) plus the pronoun היא (she / her) in its attached form.

So:

• ממני = from me
• ממך = from you
• ממנו = from him
• ממנה = from her

The doubled מ is normal in this form.

לבוא is the infinitive of to come.

The structure here is:

ביקשתי ממנה לבוא
= I asked her to come

This is a very common Hebrew pattern:

• ביקשתי ממנו לחכות = I asked him to wait
• ביקשנו מהם לעזור = we asked them to help

So after לבקש מ־מישהו, Hebrew often uses an infinitive to say what you asked that person to do.

Yes. That is also correct.

Both of these work:

• ביקשתי ממנה לבוא
• ביקשתי ממנה שתבוא

They both mean I asked her to come.

Very roughly:

• לבוא version is a little more compact
• שתבוא version is a little more explicit, because it uses a full clause: that she come / that she should come

Both are natural Hebrew.

בשבע ורבע means at seven fifteen or at a quarter past seven.

Literally:

• שבע = seven
• ורבע = and a quarter

Hebrew often tells time this way:

• שבע ורבע = 7:15
• שבע וחצי = 7:30
• שמונה פחות רבע or רבע לשמונה = 7:45

So שבע ורבע is exactly quarter past seven.

The prefix ב־ here means at.

So:

• בשבע = at seven
• בשבע ורבע = at seven-fifteen

Hebrew normally uses ב־ with clock times:

• בחמש = at five
• בשמונה = at eight
• בעשר וחצי = at ten thirty

כדי introduces a purpose: in order to, so that.

So:

• כדי שנוכל לשתות קפה = so that we can drink coffee

It explains why she was asked to come at that time.

Because the normal pattern is כדי ש־...

That ש־ means that, and it links the purpose clause to the rest of the sentence.

So:

• כדי שנוכל = so that we can
• literally: in order that we will be able to

In Hebrew, ש־ is very often attached directly to the next word:

• שנוכל
• שתבוא
• שנלך

After כדי ש־, Hebrew usually uses a future-form verb, even when English might use can, could, or so that we can.

So:

• נוכל is the future, first person plural form of יכול (to be able / can)
• שנוכל literally looks like that we will be able
• but in this kind of sentence it naturally means so that we can

This is very normal Hebrew grammar.

Because after יכול / אוכל / נוכל (can / be able to), Hebrew uses an infinitive.

So:

• נוכל לשתות = we will be able to drink / we can drink

This is the same pattern as:

• אני יכול לבוא = I can come
• הם יוכלו לעזור = they will be able to help

So לשתות is there because it follows נוכל.

לפני means before.

It is followed directly by the noun:

• לפני הפגישה = before the meeting

Notice:

• פגישה = meeting
• הפגישה = the meeting

So לפני הפגישה is literally before the meeting.

Hebrew does not need an extra word like English the before before; it just puts the definite article on the noun itself.

Because the sentence is referring to:

• a general coffee / some coffee → קפה (no ה־)
• a specific meeting → הפגישה (the meeting)

So:

• לשתות קפה = to drink coffee
• לפני הפגישה = before the meeting

This is very natural Hebrew usage.

Not here. נוכל is first person plural (we), and in this form it does not distinguish masculine vs. feminine.

So it can mean:

• we (men)
• we (women)
• we (mixed group)

The same form works for all of them.

The sentence as written is very natural, but Hebrew word order is somewhat flexible.

This version is straightforward:

• ביקשתי ממנה לבוא בשבע ורבע, כדי שנוכל לשתות קפה לפני הפגישה.

You could rearrange parts in some contexts, but this order is clear and idiomatic:

1. the request
2. the time
3. the purpose
4. the event that comes later

So for a learner, this is a very good model to follow.