المستشفى بهاي المنطقة.

A big clue is هاي for this. In Modern Standard Arabic, you would normally expect هذه instead. Also, Levantine very often uses بـ for location, where MSA would more typically use في.

A more MSA-style version would be:

المستشفى في هذه المنطقة

So المستشفى بهاي المنطقة sounds clearly colloquial/Levantine.

In Arabic, especially in the present tense, the verb to be is usually left out.

So:

المستشفى بهاي المنطقة

literally looks like the hospital in this area, but it naturally means the hospital is in this area.

If you wanted another tense, you would usually add a verb, for example:

• كان المستشفى بهاي المنطقة = The hospital was in this area
• رح يكون المستشفى بهاي المنطقة = The hospital will be in this area

بهاي breaks down into:

• بـ = in / at
• هاي = this (feminine)

So بهاي means in this.

In other words:

بهاي المنطقة = in this area

In casual writing, Levantine speakers often attach these parts together like this.

Because منطقة is a feminine noun in Arabic.

So the demonstrative has to match it:

• هاي المنطقة = this area
• not a masculine this

In Levantine, the exact form of this can vary by region, but هاي is a very common feminine form.

Yes — and that is extremely common.

بهالمنطقة means the same thing as بهاي المنطقة: in this area.

Many speakers would actually prefer:

المستشفى بهالمنطقة

in everyday speech.

So both are good, and both are natural Levantine.

In Levantine Arabic, بـ is very commonly used for location, especially in everyday speech.

Examples:

• بالبيت = at home / in the house
• بالجامعة = at the university
• بهاي المنطقة = in this area

You can sometimes hear في too, but بـ is very normal and idiomatic here.

A rough Levantine pronunciation is:

il-mustáshfa or el-mustáshfa

A few useful points:

• the الـ is usually pronounced il or el in speech, not al
• the final ى sounds like a
• the ل in الـ is pronounced here, because م is not a sun letter

So do not pronounce it like mustashfi.

Because it means the hospital, not a hospital.

So the sentence is talking about a specific hospital, one that is already known from the context.

If you wanted to say There is a hospital in this area, you would usually use an existential structure, for example:

في مستشفى بهاي المنطقة
or
في مستشفى بهالمنطقة

That changes the meaning from the hospital is in this area to there is a hospital in this area.

Yes. In Levantine, demonstratives can often come after the noun too.

So besides هاي المنطقة, you may also hear:

المنطقة هاي

Both mean this area.

With the preposition, speakers may use forms like:

• بهاي المنطقة
• بالمنطقة هاي
• بهالمنطقة

All of these are possible in Levantine, though بهالمنطقة and بهاي المنطقة are especially common.