How did Omar ibn al-Khattab send word to Amr ibn al-Aas, the governor of Egypt, during the Year of Ash, seeking aid, when Egypt had not yet been conquered?

The issue you raised regarding the date of the conquest of Egypt is correct, which led some historians to favor an earlier conquest of Egypt than the year 20 AH. Ibn al-Athir said: "It was said: in this year - meaning the year 20 - Egypt was conquered... And it was said: Egypt was conquered in the year 16 in Rabi' al-Awwal... And its conquest should have been before the Year of Ramadah, because Amr ibn al-As carried food in the Qulzum Sea from Egypt to Medina."

Ibn Kathir quoted various opinions, saying: "Muhammad ibn Ishaq said: In this year - year 20 - Egypt was conquered, and so did al-Waqidi say: It and Alexandria were conquered in this year. And Abu Ma'shar said: Egypt was conquered in the year 20, and Alexandria in the year 25. And Saif said: Egypt and Alexandria were conquered in the year 16 in Rabi' al-Awwal thereof. And Abu al-Hasan Ibn al-Athir in al-Kamil preferred that due to the story of Amr ibn al-As sending provisions from Egypt during the Year of Ramadah."

It is possible that the Year of Ramadah was later than the year 18. Ibn Kathir mentioned that the Plague of Amwas and the Year of Ramadah were in the year 18, then he mentioned that Umar ibn al-Khattab wrote to Amr ibn al-As in Egypt asking for aid for the Ummah of Muhammad, so he sent provisions to him.

However, this is countered by the fact that Egypt had not yet been conquered in the year 18, which means that the Year of Ramadah was either after the year 18, or the mention of Amr ibn al-As in the context of the Year of Ramadah is an error.