اليوم أقرأ بريدا إلكترونيا من صديقي في المكتب.

Yes. Arabic word order is fairly flexible, especially with time expressions.

• اليوم أقرأ بريدا إلكترونيا... = Today, I am reading an email...
• أقرأ اليوم بريدا إلكترونيا... = I am reading an email today...

Starting with اليوم gives today a little extra prominence. Even though the sentence begins with a time word, it is still basically built around the verb أقرأ.

Because اليوم can function by itself as a time adverb, meaning today.

Arabic often uses bare time words without a preposition:

• اليوم = today
• غدًا = tomorrow
• أمس = yesterday

Grammatically, اليوم here is an adverbial expression of time (ظرف زمان).

In MSA, أقرأ is the imperfect form, and it can mean either:

• I read
• I am reading

Context decides which English translation sounds best. In this sentence, with اليوم, English often prefers I am reading or I read today, depending on the context.

Because the word contains two different hamzas:

• the first أ is the I- prefix of the imperfect verb
• the final ء belongs to the root قرأ = to read

So أقرأ is pronounced roughly aqraʾu in careful MSA pronunciation, or aqraʾ in pause.

Because it is a noun + adjective combination:

• بريد = mail/post
• إلكتروني = electronic

So literally it means electronic mail. In modern usage, this can mean email. If you want to stress one email message, Arabic also often uses رسالة إلكترونية.

Because in Arabic, adjectives normally come after the nouns they describe.

So:

• بريد إلكتروني = electronic mail
• literally: mail electronic

This is the normal pattern in Arabic, unlike English.

Because both words are indefinite accusative here.

• بريدا is the direct object of أقرأ, so it is in the accusative
• إلكترونيا is an adjective describing بريدا, so it must match it

Arabic adjectives agree with the noun they describe in:

• definiteness
• gender
• number
• case

So both appear in the same form: بريدًا إلكترونيًّا in fully vowelled writing.

صديقي is made of:

• صديق = friend
• ـي = my

So صديقي means my friend.

The preposition من means from, so:

• من صديقي = from my friend

Also, prepositions in Arabic make the following noun genitive. With صديقي, the possessive ending ـي is attached, so the case ending is not separately obvious in the spelling.

Because المكتب means the office, while مكتب would mean an office or office in a more general sense.

So:

• في المكتب = in the office
• في مكتب = in an office

Using ال suggests a specific office, usually one understood from the context.

It can be understood either way unless context makes it clear.

It could mean:

• I am reading an email in the office
• I am reading an email from my friend who is in the office

This kind of attachment ambiguity is normal in Arabic, just as it can be in English. In real use, context usually tells you which meaning is intended.

A fully vowelled version is:

اليومَ أقرأُ بريدًا إلكترونيًّا من صديقي في المكتبِ.

Roughly pronounced:

al-yawma aqraʾu barīdan iliktrūniyyan min ṣadīqī fī al-maktabi

A few notes:

• اليومَ is accusative as a time expression
• أقرأُ ends in -u in careful MSA
• بريدًا إلكترونيًّا is accusative because it is the object
• المكتبِ is genitive after في

In normal pauses and less formal pronunciation, some final short vowels may be reduced or not clearly pronounced.