По приезде в центр мы встретились с подругой у гостиницы.

It means upon arrival or after arriving.

In this sentence, по приезде в центр means when/after we arrived in the center. It is a compact, slightly formal way to express time.

A more expanded version would be:

• Когда мы приехали в центр, мы встретились с подругой у гостиницы.

So по приезде is not the main action itself; it sets the time for the main action встретились.

Because по приезде is a fixed standard expression where по is followed by the prepositional case, not the dative.

This is one of those patterns learners have to memorize:

• по окончании = after the end/completion
• по прибытии = upon arrival
• по приезде = upon arrival

You may hear по приезду in colloquial speech, but standard written Russian strongly prefers по приезде.

It is perfectly correct and common, but it sounds a bit more compact and formal than everyday speech.

In everyday conversation, many speakers would more naturally say:

• Когда мы приехали в центр...
• После того как мы приехали в центр...
• После приезда в центр...

So по приезде is natural, but a little more written-style or formal than a full clause with когда.

Because в центр shows movement toward a destination, so Russian uses the accusative case.

• в центр = into/to the center
• в центре = in the center

Compare:

• Мы приехали в центр. = We arrived in/went to the center.
• Мы были в центре. = We were in the center.

So here the idea is arrival to a place, not being located there.

Встретились is perfective, so it presents the meeting as a single completed event.

• встретились = met, got together, met up
• встречались = were meeting / used to meet / met repeatedly, depending on context

In this sentence, the speaker is talking about one completed meeting after arriving in the center, so встретились is the natural choice.

These two patterns mean different things.

• встретиться с кем-то = to meet up with someone
• встретить кого-то = to meet/come across someone

So:

• Мы встретились с подругой = We met up with a female friend.
• Мы встретили подругу = We encountered/saw the female friend.

The version with -ся often suggests a mutual meeting or an arranged meeting. That is why it fits well here.

The -сь is the reflexive ending of the verb встретиться.

Here it does not mean the literal English idea of oneself. Instead, it is part of the verb and helps create the meaning to meet each other / to meet up.

Compare:

• встретить = to meet, to encounter
• встретиться = to meet up, to meet one another

So встретились means we met up.

Подругой is the instrumental singular of подруга.

The verb встретиться normally takes:

• с + instrumental

So:

• встретиться с другом
• встретиться с подругой
• встретиться с коллегами

That is just the standard government pattern of the verb.

Also, подруга usually means female friend, not necessarily girlfriend in the romantic sense. If the speaker means a romantic girlfriend, context is important; often Russian would use девушка for that meaning.

Because у гостиницы means by / near / outside the hotel, while в гостинице means in the hotel.

So:

• у гостиницы = at the hotel, by the hotel, near the hotel
• в гостинице = inside the hotel

In your sentence, the meeting point is near the hotel rather than inside it.

It is the genitive singular.

The preposition у requires the genitive:

• у дома
• у метро
• у гостиницы

So the dictionary form гостиница changes to гостиницы after у.

Yes, Russian word order is flexible.

This sentence begins with По приезде в центр because that time phrase sets the scene first: upon arriving in the center...

Other word orders are possible, for example:

• Мы встретились с подругой у гостиницы по приезде в центр.
• С подругой мы встретились у гостиницы по приезде в центр.

But the original order sounds natural because it gives the time/background first and then the main event.

Yes. После приезда в центр is also correct and means nearly the same thing.

Compare the tone:

• по приезде в центр = compact, slightly more formal/written
• после приезда в центр = also correct, often a bit more neutral
• когда мы приехали в центр = very clear, everyday spoken style

So all three can work, but they differ a little in style.

Because Russian has no articles.

English needs words like a, an, and the, but Russian usually leaves that information to context.

So:

• в центр could be to the center
• с подругой could be with a friend / with the friend, depending on context
• у гостиницы could be by the hotel

The exact English article depends on the situation, not on a separate Russian word.