What is the procedure for change of name in the visa form after the petition is filed?

My h1b visa was initiated earlier this year, and it has been selected in the lottery. Assuming my petition is approved, this is my query,

I do not have a surname in my passport, hence have filled the visa form as
Given Name: FNU
Surname: XXX YYY ZZZ

I came to know that there would be problems in the US without a surname in the passport.
The US consulate website instructed to add the surname in the passport, but in my case since my petition has already been filed, is it advisable to add the surname in the passport?.
Some people said that Amendment should be filed, what is it and who how to do it?. employee or employer?

Here is the process:

  1. Get passport updated with surname

  2. Once H-1B is approved, ask employer to file H-1B amendment to get the name updated to match passport name. This would take few months to process.

  3. Once H-1B amendment has been approved, go for visa stamping and get the visa stamp in updated name