First, the whole reason that Ishmael was sent away was because he was making fun of Isaac on the day Isaac was weaned.
Genesis 21:8-21As for the question of how old Ishmael was at that time, the Bible does not say precisely. We know that Abraham had Ishmael when Abraham was 86 (Genesis 16:16), and that Abraham had Isaac when he was 100 (Genesis 21:5). Thus, Ishmael was at least 14 when Isaac was weaned, and perhaps older (maybe 17 or 20). We know that Sarah died at 127 (Genesis 23:1), and that she had Isaac when she was 90 (Genesis 17:17), which gives a maximum age of 51 for Ishmael.
And the child grew, and was weaned: and Abraham made a great feast the same day that Isaac was weaned. And Sarah saw the son of Hagar the Egyptian, which she had born unto Abraham, mocking. Wherefore she said unto Abraham, "Cast out this bondwoman and her son: for the son of this bondwoman shall not be heir with my son, even with Isaac."
And the thing was very grievous in Abraham's sight because of his son.
And God said unto Abraham, "Let it not be grievous in thy sight because of the lad, and because of thy bondwoman; in all that Sarah hath said unto thee, hearken unto her voice; for in Isaac shall thy seed be called. And also of the son of the bondwoman will I make a nation, because he is thy seed."
And Abraham rose up early in the morning, and took bread, and a bottle of water, and gave it unto Hagar, putting it on her shoulder, and the child, and sent her away: and she departed, and wandered in the wilderness of Beersheba. And the water was spent in the bottle, and she cast the child under one of the shrubs. And she went, and sat her down over against him a good way off, as it were a bow shot: for she said, Let me not see the death of the child. And she sat over against him, and lift up her voice, and wept.
And God heard the voice of the lad; and the angel of God called to Hagar out of heaven, and said unto her, "What aileth thee, Hagar? fear not; for God hath heard the voice of the lad where he is. Arise, lift up the lad, and hold him in thine hand; for I will make him a great nation."
And God opened her eyes, and she saw a well of water; and she went, and filled the bottle with water, and gave the lad drink. And God was with the lad; and he grew, and dwelt in the wilderness, and became an archer. And he dwelt in the wilderness of Paran: and his mother took him a wife out of the land of Egypt.
The text, however, makes clear that Ishmael had not yet married and repeatedly refers to him as a child, suggesting that he was at the younger end of the spectrum (from age 14-17 or so).
Zakir seems to think that the text says that the child Ishmael was placed on Hagar's shoulder. In the Authorized Version, what is on her shoulder is a water container (presumably an animal skin full of water), called a "bottle of water." In the NASB, it is both the water and bread on her shoulder.
This was the way that women of that time carried water, as can be seen in Rebekah doing the same thing (Genesis 24:15 and 45 and 24:46).
By contrast, nowhere in the books of Moses are women described as carrying young children on their shoulders (to my knowledge). Indeed, given that Hagar is already carrying water on one of her shoulders (and perhaps bread also?), it would be a little odd for her to try to simultaneously carry a child on the other shoulder.
In short, there is no good reason to suppose that Ishmael was a mere infant carried on Hagar's shoulders, when she was sent away.
Second, Hagar was not sent to Arabia. She was just sent away. For her, that "away" was toward her homeland of Egypt. The wilderness of Paran is somewhere between Israel and Egypt. More particularly it seems to be between Midian and Egypt (see 1 Kings 11:18), probably on the Sinai peninsula. It was somewhat close to Egypt, which is where Hagar found Ishmael's wife (see this map as one example of the approximate location of Paran).
Thus, contrary to Zakir's assertions, there is no contradiction in the text. Ishmael was a young boy, but not an infant, when he was sent away. He almost died of thirst in the wilderness when the water ran dry, but God miraculously saved him and his mother and made a great nation of him.