The completed sentence is:

"The Emir and conqueror of the enemy territories arrives next week"
The subject "The Emir and conqueror of the enemy territories" refers to a single person, so the verb must be singular. "Arrives" is the correct singular form of the verb for this context.

It is important to note the distinction between the sentences:

1. The Emir and conqueror of the enemy territories arrives next week

2. The Emir and the conqueror of the enemy territories arrive next week.

While the article "the" is absent before the second subject in no (1), it is present in no (2). The use of the "the" article  in no (2) helps to show that "the Emir" and "the conqueror of the enemy territories" are two different people. So in that case, the verb that agrees with the subject is a plural verb.