Learn Hardcore Hungarian: A telefonon új üzenet van, ezért felhívom a barátnőmet. - There is a new message on the phone, so I will call my girlfriend.

-ban/-ben (inessive) = “in” something (physical interior), e.g., a táskában “in the bag”
-ra/-re (sublative) = “onto/to” a surface/destination, e.g., az asztalra “onto the table”

So a message is considered to be “on” the phone: a telefonon.

Should it be a telefonomon (“on my phone”) instead of a telefonon?

If you mean specifically “on my phone,” use the possessed form: a telefonomon (“on my phone”). The given a telefonon means “on the phone” as a specific device already mentioned or contextually known. Both are grammatical; the possessed form is more explicit about whose phone it is.

Why is van used here? I thought Hungarian often drops “to be” in the present.

Hungarian drops present-tense van only in predicate nominal/adjectival sentences:

Az üzenet új. “The message is new.” (no van)

But in existential/locative sentences (“there is/are” somewhere), van is required:

A telefonon új üzenet van. “There is a new message on the phone.”

So you must keep van here.

Why isn’t there an indefinite article egy before új üzenet?

In existential statements, Hungarian often omits egy when simply stating existence:

A telefonon új üzenet van.

Adding egy is possible and adds a light emphasis to “one/a single” or introduces it as a notable new item:

A telefonon egy új üzenet van.

Both are correct; without egy is the more neutral “there is (some) new message.”

Can I change the word order? For example: Van új üzenet a telefonon or A telefonon van új üzenet?

Yes, and each order tweaks the emphasis (focus is immediately before the verb):

A telefonon új üzenet van. Topic = “on the phone”; focus = “new message.”
Van új üzenet a telefonon. Neutral “There is a new message on the phone.”
A telefonon van új üzenet. Emphasizes the location: “It’s on the phone (as opposed to somewhere else) that there is a new message.”

All are grammatical; choose based on what you want to highlight.

What exactly does ezért mean, and why is there a comma before it?

Ezért means “therefore/for this reason,” linking cause and result. In Hungarian, a comma typically precedes such linking elements between independent clauses, so the comma before ezért is standard:

…, ezért … = “..., therefore ...”

Could I use emiatt, szóval, or úgyhogy instead of ezért?

emiatt = “because of this/for this reason” (cause adverbial): Emiatt felhívom… (slightly more formal than ezért).
szóval = “so/so then” (discourse marker; informal, conversational).
úgyhogy = “so/so that” in the sense of “as a result” (very colloquial; often no comma).

All can work, but register and punctuation vary. Ezért is a clean, neutral connector.

Why felhívom and not just hívom?

The verb hív means “to call” in general (call by name, invite). Felhív specifically means “to call on the phone.” So for phoning someone, use felhív:

Hívom Pétert. could mean “I’m calling Peter (over)” or “I’m calling him (by name).”
Felhívom Pétert. clearly “I’m calling Peter on the phone.”

Why does the verb end in -om (felhívom) instead of -ok (felhívok)?

Hungarian has definite vs. indefinite conjugation:

Use the definite ending when the direct object is definite/specific (proper name, definite article, possessed noun, demonstrative, etc.): felhívom a barátnőmet.
Use the indefinite ending when the object is indefinite/unspecified: felhívok egy barátnőt (“I’ll call a (female) friend”).

Here a barátnőmet is definite, so felhívom is required.

What does the -met in barátnőmet show?

It combines possession and the accusative:

Base: barátnő “girlfriend/female friend”
1st-person possessed: barátnőm “my girlfriend”
Accusative of possessed noun: barátnőmet “my girlfriend” (as object)

Note the linking vowel -e- before -t. Also, possessed nouns normally take the article in Hungarian: a barátnőm / a barátnőmet.

Is barátnő always “girlfriend”? What if I just mean a female friend?

Context usually makes barátnő mean “girlfriend.” To avoid that and mean “female friend,” you can say:

nőbarátom or lánybarátom (clearer but less common) Colloquial for “girlfriend”: a csajom, neutral/partner: a párom. In many contexts, a barátnőm will be understood as “my girlfriend.”

Where does the verbal prefix fel- go if I start with ezért or in other structures?

Default (neutral) order is prefix + verb: felhívom. The prefix moves after the verb when there is focus, negation, question words, or in some imperative/subordinate contexts:

Most hívom fel a barátnőmet. (focus on “now”)
Nem hívom fel. (negation)
Kit hívsz fel? (question)

With sentence-initial ezért, both are heard:

Ezért felhívom a barátnőmet. (neutral)
Ezért hívom fel a barátnőmet. (slight emphasis on the calling action) Both are acceptable in practice.

How do I say this in the future?

Hungarian often uses the present for near-future plans:

Ezért felhívom a barátnőmet. “So I’ll call my girlfriend.”

To make the future explicit, use fog + infinitive:

Ezért fel fogom hívni a barátnőmet.

Both are correct; the present is very common for planned/near-future actions.

Can I drop the article and say Telefonon új üzenet van?

Better keep the article. Telefonon without an article is typically the adverbial “by phone” (as a means), e.g., telefonon beszélünk “we talk by phone.” To refer to a specific device as a location, use the article: A telefonon új üzenet van.

Are there other natural ways to say “There’s a new message on my phone”?

Yes:

A telefonomon van egy új üzenet.
Van egy új üzenetem a telefonomon. (“I have a new message on my phone,” using possession) All are idiomatic; choose based on what you want to emphasize (existence vs. possession vs. location).

Any tips on pronouncing the long vowels (í, ő, ó) in this sentence?

í (in hívom) is a long “ee” sound.
ó (in hívom) is a long “o” like in “more” but without the English glide.
ő (in barátnőmet) is a long front rounded vowel; round your lips as for “o” but keep the tongue position closer to “e.”

Rough syllable guide: A te-le-fo-non új ü-ze-net van, e-zért fel-hí-vom a ba-rát-nő-met.