On the morrow (tēi epaurion). Locative case of article with the compound adverb (hēmerāi day being understood), the second day after leaving Caesarea, 28 miles from Joppa. The third day (the next morrow, Act 10:23) they start back home and the fourth day (on the morrow again, Act 10:24) they reach Caesarea.
As they (ekeinōn). The party of three from Caesarea. Genitive absolute with present participle hodoiporountōn (journeying) and eggizontōn (drew nigh).
The housetop (to dōma). Old word and in Gospels (Luk 3:19, etc.), but only here in Acts. From demō, to build, and so any part of the building (hall, dining room, and then roof). The roof was nearly flat with walls around and so was a good place for meditation and prayer and naps.