1) Change to this :
@using (Ajax.BeginForm("Edit", "Cars",null, new AjaxOptions
{
InsertionMode = InsertionMode.Replace,
HttpMethod = "POST",
OnSuccess = "updateSuccess"
}, new { @id = "updateCarForm" }))
-- You forgot to overload the method.
2) Your form is reloading(redirecting) because you have not included
jquery.unobtrusive-ajax.min.js
file into your file. download and then it should work as axpected.