Often the term foreshadowing is used of negative things, such as death, that are going to happen in a story. In the case of C.S. Lewis, he foreshadows many things, good and bad. Lewis is such a masterful storyteller that he plants many small seeds throughout a story, which later blossom into significance.
In The Horse and His Boy, there are many instances of foreshadowing of events big and small. I will discuss the foreshadowing of two major themes: the mystery of Shasta's origin, and the involvement of Aslan.
When Shasta is first introduced, it is this way:
... far south in Calormen on a little creek of the sea, there lived a poor fisherman called Arsheesh, and with him there lived a boy who called him Father.
Thus, right away we are notified that Shasta is apparently not Arsheesh's real son, but thinks he is.
Later in the same chapter, a visitor to Arsheesh points out (in a conversation on which Shasta eavesdrops) that Shasta cannot possibly be Arsheesh's son, because unlike Arsheesh he is fair-skinned and blue-eyed. Arsheesh then admits that he got Shasta from a boat that washed up on the shore, containing only a baby (Shasta) and a man who had apparently starved himself to keep the baby alive.
Shasta is excited to hear that he is not Arsheesh's son:
"Why, I might be anyone!" he thought. "I might be the son of a Tarkaan myself -- or the son of the Tisroc (may he live for ever) -- or of a god!"
These dreams of Shasta's turn out to be surprisingly accurate, though it will take almost the whole book before he finds out that he is actually of royal blood.
Later, when Shasta, Aravis, and the two talking horses are trying to sneak through the city of Tashbaan (Chapters 4 and 5), Shasta is mistaken for a Northern prince by some Narnia lords who are visiting the city. Apparently, Prince Corin has run away, and the Narnians have been looking for him, and Shasta looks just like him. They grab him, calling him Corin, and drag him back to their quarters, scolding him for the worry he has caused them. In this way, he ends up hearing their plans. Later, when they conclude that he has sunstroke and leave him alone to rest, Shasta encounters the real prince Corin, who is returning from his escapades in the city. The boys do not realize it, but they are identical twins. Corin's parting words to Shasta are, "I hope we meet in Archenland. Go to my father King Lune and tell him you're a friend of mine."
Of course, Shasta will end up encountering King Lune when he speeds North with a desperate warning that the Calormen prince is planning an attack on Archenland. Lune will recognize Shasta as his long-lost son, the heir to the throne.
Besides the foreshadowing about Shasta's birth (which is a major plot point), there is, as with every book in the Narnia series, plenty of foreshadowing about the great lion Aslan before the characters actually meet him.
The characters encounter lions when escaping by night from Calormen, and it is in trying to get away from the lions that Shasta and the horse Bree first meet up with Aravis and the talking horse Hwin.
Later, when Shasta is waiting for his friends by the tombs outside Tashbaan, he is comforted by a large tabby cat. In the middle of the night this cat drives off jackals that are threatening Shasta. When he first sees it doing this, it looks so big that Shasta takes it for a lion.
Later still, when the horses are racing to bring their warning of a surprise attack to King Lune, a lion chases them (causing them to run much faster) and rakes its claws across Aravis's back. Shasta charges it to shoo it away, and to his surprise it goes.
All these are small incidents (as foreshadowing often is), and do not stick with the reader for long because they are always swept away by a new and exciting plot development. But when Shasta does finally meet Aslan, Aslan informs him that "I was the lion."
"I was the lion who forced you to join with Aravis. I was the cat who comforted you among the houses of the dead. I was the lion who drove the jackals from you while you slept. I was the lion who gave the horses the new strength of fear for the last mile so that you should reach King Lune in time. And I was the lion you do not remember who pushed the boat in which you lay, a child near death, so that it came to shore where a man sat, wakeful at midnight, to receive you."
The idea that "Aslan is at the back of all the stories" is a major theme of the whole Narnia series.