Its been a while so some of the numbers maybe incorrect. It is a pilot explanation not a detailed technical explanation.
The IRS does not update position but the FMC using IRS and external inputs does update its position. The FMC navigates the track.
Without GPS the FMC updates position usually by DME/DME/IRU. After 12 minutes without FMC DME/DME/IRU updating the message IRS NAV ONLY appears.
The FMC begins to apply a 'worst case' IRS drift allowance of 0.833 NM/hour to the ANP, so after about 2.3 hours without a DME/DME/IRU update (or LOC/DME, VOR/DME if enabled in your aircraft) you will see the message 'unable reqd nav perf RNP'. This does not actually mean the aircraft is actually off track, it just means that the confidence of the nav solution is outside calculated statistical probability of position. Note this is based on the worst case drift allowance (I was told that it was a rate based of errors in an iron gyro system applied to a ring laser gyro for certification but don't have any documentation to confirm). Ring laser gyro usually performs much better than that which is why you only see small map shifts when the FMC is eventually updated with a good DME/DME update. The ANP will rapidly decrease once a good position update is obtained.
Single navaid destinations were always problematic as the position update was never very accurate, and I recall that there was an altitude and distance limit before an update occurred (15,000 feet and 25 NM are numbers I recall but it may be in error after many years out of the B737).