Will 16 mg of codeine (over the counter pain med from ireland) taken saturday and sunday show up on a preemployment drug test on wed morning?

unlikely: If you only ingested what you have said, it is likely that your urine test will have dropped below the cutoff levels and be reported negative. This is no guarantee however and if the medication was obtained over the counter without prescription in a foreign country, the medical review officer will likely not accept that as medical justification and be required to report that as a positive test.