Ptolemais

How far was "Ptolemais" (Acts 21:7) from Tyre?
Ptolemais was only about 25 miles (40 kilometers) south of Tyre, toward Jerusalem.

Wasn't Paul "hurrying to be at Jerusalem ... on the Day of Pentecost"  (Acts 20:16)?
 

Then why didn't they just walk to Ptolemais, which would have taken only one day, instead of waiting for seven days in Tyre for the ship?
Pentecost is fifty days after the Passover, which Paul celebrated in Philippi just prior to starting his sea voyage toward Israel (see Acts 20). He probably felt comfortable about arriving in Jerusalem on time since it is only 120 miles (193 kilometers) south of Tyre, and the less predictable 900 mile (1,450 kilometer) sea voyage from Philippi to Tyre had been covered in 30 days:
 
5 days from Philippi to Troas
7 days in Troas (Acts 20:6)
1 day (estimated) from Troas to Assos (Acts 20:13)
1 day (estimated) from Assos to Mitylene (Acts 20:14)
1 day (estimated) from Mitylene to Chios (Acts 20:15)
1 day from Chios to Samos/Trogyllium (Acts 20:15)
1 day from Samos/Trogyllium to Miletus (Acts 20:15)
1.5 days (est.) for a messenger to travel from Miletus to Ephesus (Acts 20:17)
1.5 days (est.) for the elders to travel from Ephesus to Miletus (Acts 20:18)
1 day (estimated) spent with the Ephesian elders in Miletus (Acts 20:18-38)
1 day (estimated) from Miletus to Cos (Acts 21:1)
1 day from Cos to Rhodes (Acts 21:1)
1 day (estimated) from Rhodes to Patara (Acts 21:1)
6 days (estimated) from Patara to Tyre (Acts 21:2-3)
30 days total

How can the 400 miles from Patara to Tyre be sailed in 6 days when it took 5 days to sail the 150 miles from Philippi to Troas?
While the crossing from Philippi to Troas took 5 days this time (see Acts 20:6), Paul's first crossing of the same 150 miles in the opposite direction from Troas to Philippi/Neapolis took only 2 days (75 miles/day), including an overnight at Samothrace (Acts 16:11). The distance sailed depended on the direction of the wind, which appears to have been favorable since Paul's ship covered about 50 miles per day during the island hopping phase of his journey. The second ship sailed directly from Patara to Tyre. Since it didn't make overnight stops at ports, it would have easily averaged 67 miles (108 kilometers) per day to cover the 400 miles in 6 days.

Where is "Caesarea" (Acts 21:8)?
The capital of the Roman province of Judea, Caesarea is 30 miles (48 kilometers) south of Ptolemais.

Who went to Caesarea "on the next day" (Acts 21:8)?
Luke wrote, "we who were Paul's companions" (Acts 21:8), so those accompanying Paul, including Luke. Since they stayed "many days" (Acts 21:10), Paul most likely joined them later.

Who is "Philip" (Acts 21:8) who had the "four virgin daughters who prophesied" (Acts 21:9)?
"One of the seven" (Acts 21:8) deacons originally chosen by the church in Jerusalem (see Seven Men).

Why is Philip called "the evangelist" (Acts 21:8)?
He evangelized the Samaritans (see Samaria), as well as the Ethiopian eunuch (see Philip and the Ethiopian eunuch), and likely had continued to evangelize during the twenty or so years that had passed since then.