I was a qualified Officer of the Deck in the US Navy. I have around 2000 hours watchstanding time and about 50 port and anchorage transits.
A (competent) 1500s sea captain's navigational skills would be much higher than any modern day sailor's, just due to the lack of technology at the time. Modern navigation is, of course, done by computer. You will plot your navigational position by GPS and soundings (depth readings) by fathometer. Only in the last 5 to 10 years, these technologies have become advanced enough that you no longer need to rely on older methods.
That being said, while I'm certain the modern merchant fleets of Maersk and CMA CGM are using all-digital navigation, there are likely a lot of smaller local transports still using older methods. In particular, I know that the majority of non-oil tankers in West Africa are pretty backwards, and a surprising amount of the merchant traffic in the Mediterranean is smaller local merchants.
For ships without the newest technology, a lot of what the old sea captain knows would be very valuable. Taking fixes by compass upon entering port is unchanged from the 1500s. Where GPS is not available or broken, the sextant must be used to take fixes from the stars in open ocean. The sextant requires a lot of skill to use; the old mariner's skill with the astrolabe would translate directly with a little bit of training. I can attest that on a modern US Navy Warship in 2010, my captain launched a small boat to sound a harbor to verify depth before we tried to enter. A 1500s sea captain would have forgotten more about this technique than my entire ship's company ever knew. Finally, general knowledge of wind and waves and how the sea looks can be useful in a lot of small ways. I knew a lot more about currents just looking at the ocean's surface after 2000 hours than I did when I started. A 1500s sea captain would know far more still.
I imagine there are still a lot of local merchants that don't use electronic charts. In that case, while the old mariner would be amazed by modern chart's accuracy, using them is not fundamentally different from the portolans that he would have been familiar with.
In conclusion, for not state-of-the-art vessels, a 1500s ship captain would have a lot of usable skills that could be upgraded with minimal training. Remember, things have gotten easier in the last 500 years, not harder. This old mariner learned the hard way.
Not prone to sea sickness
It is somewhat amazing how many career US Navy sailors are still susceptible to seasickness after a decade or more in the Navy. Modern ships tend to be very large and stable, but then once in while you are in 20 foot seas and you just have to puke. But seasickness is something you get used to and get over. A 1500s sea captain would presumably be quite immune. Not a huge advantage, but useful.
A modern US Navy Boatswain's Mate (pronounced Bosun's mate) has a knowledge base that is largely unchanged for the last 3000 years. How to tie knots in ropes, how to unwind the strands of a rope and remake it into an eye, how to tie things down so they don't fall overboard at sea; these skills remain.
Its actually an open question whether a 1500s sea captain would have these skills. For example, Horatio Nelson probably never knew of these things. He was an aristrocrat and learned navigation and the stars and how to maneuver a ship. There were career enlisted sailors (namely, the ship's bosun) that knew these things. Depending on what your 1500s captain did before he was a captain, he may know more or less about these things.
The skills and general acclimation with the sea would make the 1500s sea captain a very useful person to have around. He would need some training with modern equipment, with what GPS is, with modern techniques of marking transit of charts. He would also need to learn from scratch how to direct a ship under power, as opposed to under sail.
But given his skill set, it wouldn't be a stretch to hire such a person as a third mate on a 3rd world coast-hugging cargo tramp.