Congratulations on making silver.

You should be fine by then. I seem to think mine took about 5 days or so to update online which is really the thing that matters. You won't need the physical card which might take longer to arrive.

Mind you, having just returned from LAS on Friday, I'd encourage you to minimise time at the airport rather than plan to relax in the lounge. The airport is so close to the Strip that you won't need much contingency for traffic etc and the lounge is a) nothing special and b) shared so it was really busy.