How do I shift the elements inside a Rust vector to the right and put the out-of-bounds element at the beginning?

You are looking for [T]::rotate_right and [T]::rotate_left. Example (Playground):

let mut v = vec![1, 2, 3, 4, 5];
v.rotate_right(1);
println!("{:?}", v);

This outputs:

[5, 1, 2, 3, 4]

If you find yourself calling rotate_* a lot, you should consider using a different data structures, as those methods are linear time operations. See this answer, for example.

VecDeque is a collection similar to a Vec but optimized for adding/removing elements at either end. Just like slices, it has rotate_{left,right} methods, but those are more efficient than Vec's (O(min(mid, len() - mid)) time for VecDeque, vs. O(len()) time for Vec):

use std::collections::VecDeque;

fn main() {
    let mut v = (1..6).collect::<VecDeque<_>>();
    v.rotate_right(1);
    println!("{:?}", v);
}

(Permalink to the playground)