Paying Taxes on a Bonus Check

Can an employer just put $500 into my 401k to avoid taxes and the IRS is OK with that?

You can typically designate a portion of bonuses and other irregular income to go to your 401(k), which is perfectly acceptable so long as you don't exceed annual contribution maximums. I don't know if you'll be able to do it after the bonus is paid out, though. Check with your benefits coordinator to see what options you have.

If not, then is putting $390 ($500 * 22% supplemental income) into a backdoor Roth the only way to save this bonus?

I'm not sure what you mean by backdoor Roth here - effectively you're having 22% withheld and can deposit the rest in a Roth (subject to all other contribution requirements). There's not really any immediate tax savings here.

Note that the 22% is just an estimate of the tax that will be due. When you file, the bonus will all be considered income (unless you designate some of it to your 404(k) as described above) and will effectively be taxed at your marginal rate, which may be higher than 22%.

Can an employer just put $500 into my 401k to avoid taxes and the IRS is OK with that?

The company I work for, and the company my spouse works for, put the profit sharing directly into the 401(k). It counts as a company contribution not as part of the employee contribution. There are zero income taxes, and zero FICA removed from the contribution.

If not, then is putting $390 ($500 * 22% supplemental income) into a backdoor Roth the only way to save this bonus?

That would not save any taxes, because Roth money is always after tax money.

Making adjustments to withholding doesn't save you any money in the end when it comes to lump sum bonuses, it just changes the path to the same final amount when you file in Spring.