However, depending on the type of grass there is no real guarantee that the grass will die. If you just put down mulch the grass will come through it for sure. I would recommend spraying the whole strip with roundup first, wait to make sure the grass is starting to die, then spot spray any remaining green patches, put down newspaper, then put down mulch.
About your planting holes, though it is always better to amend the entire garden and not just the holes, if these are just easy to grow shrubs or tough perennials like daylilies then adding some good compost to just the holes should be alright.
P.S. - Doing things "easy" now might lead to more work cleaning up the grass or even redoing the whole thing later. Do a new garden right the first time and it can save a lot of time in the long run.