grep lines starting with "1" in Ubuntu

Your regular expression doesn't mean what you think it does. It matches all lines starting (^) with one (1) repeated zero or more (*) times. All strings match that regular expression. grep '^1' does what you want.

Did you try the following?

 ls -1 | grep "^1"

That is, remove the *, which basically tells grep, find zero or more occurances of the ^1 expression. In other words: match the lines that start with a 1, or not.

Although this doesn't answer your question, this is a better solution to what appears to be your goal:

ls -ld 1*

You can use a shell glob to list all files that start with 1. Note that * has a different meaning in shell globbing than regular expressions.