Signified (εσημαινεν). Imperfect active in Westcott and Hort, but aorist active εσημανεν in the margin. The verb is an old one from σημα (σημειον) a sign (cf. the symbolic sign in Acts 21:11). Here Agabus (also in Acts 21:10) does predict a famine through the Holy Spirit.
Should be (μελλειν εσεσθα). Μελλω occurs either with the present infinitive (Acts 16:27), the aorist infinitive (Acts 12:6), or the future as here and Acts 24:15; Acts 27:10.
Over all the world (εφ' ολην την οικουμενην). Over all the inhabited earth (γην, understood). Probably a common hyperbole for the Roman empire as in Luke 2:1. Josephus (Ant. VIII. 13, 4) appears to restrict it to Palestine.
In the days of Claudius (επ Κλαυδιου). He was Roman Emperor A.D. 41-44. The Roman writers (Suetonius, Dio Cassius, Tacitus) all tell of dearths (assiduae sterilitates) during the brief reign of Claudius who was preceded by Caligula and followed by Nero.