John Gill Commentary 1 Kings 1:35

"Then ye shall come up after him, and he shall come and sit upon my throne; for he shall be king in my stead; and I have appointed him to be prince over Israel and over Judah." — 1 Kings 1:35 (ASV)

Then you shall come up after him
When anointed, proclaimed, and cheered, then he was to mount the mule, and ride before them as their king, at the head of them; they following after, in token of their subjection to him:

that he may come and sit upon my throne ;
at Jerusalem, in the king's palace, and there exercise his kingly power he would now be invested with:

for he shall be king in my stead ;
even during David's life, as well as after his decease:

and I have appointed him to be ruler over Israel, and over Judah ;
that is, over all the twelve tribes of Israel Judah may be particularly mentioned, though included in Israel, because Adonijah had invited the men of Judah to his feast and party, (1 Kings 1:9) ; and therefore had they not been named, might think he had no power over them.