Basic answer...not taking into account axial tilt and sun details etc. Just basic climatology.
500 mile width. On Earth the distance between each degree of latitude is about 69miles. So on Earth, your landmass would average 3.5-4 degrees latitude either side of equator. On your larger planet, it would be less!
Coriolis at the equator
Earth rotates west to east, this rotation causes surface waters and winds to flow east to west.
Coriolis causes surface waters and winds to be deflected to the right in the northern hemisphere and left in the southern.
BUT, at the equator the is little to no deflection. We call it the 'doldrums'. So your coastal water currents will probably be fairly weak (if no other drivers can be found).
As mentioned by Palarran, there will be no complete ocean circulation. On Earth, all our oceans are connected and combines to create the thermohaline circulation. This transfers cold polar water to the equator and beyond and warm equatorial water to the poles. But I don't see why you can't have two separate thermohaline circulation systems.
Your planet has ocean from the poles to pretty much the equator (barring a few degrees). Your cold polar waters will extend alot further inwards than on Earth. Eg in the northern hemisphere the cold oceans are mostly barred by eurasia and north America. In the south, the cold currents have near free reign and have formed a large antarctic circumpolar current.
You will have two of these, North and south. But I believe it may be 'wider'. This will result in larger ice caps affecting air temperatures. The cold water will sink until it hits the ocean surface and then travel underwater towards the equator (you will need some sunken landmass to direct it there). When it reaches the equator it may rise and bring nutrients to the surface. See areas of upwelling on Earth.
Depending on how wide a continental shelf you have, your coastal waters will therefore be colder than expected. Shallow coast, weaker and warmer current; steeper shelf, stronger current influenced by the upwelling cold current.
Such a strong polar ocean current would also result in very little surface mixing with the water in the mid-latitudes. So thinking the problem through, you will have a excessively strong polar current and a very weak equatorial current. You may have a narrow third current between the two. A mid-latitude current. Where the equatorial and polar currents interact. I'd imagine it would be mainly driven by any atmospheric circulation as there will be no landmass to drive or focus the direction.
I don't think it will be similar to Earth's hadley-ferrel-polar cell circulation. If it is, it will be a mutated version.
You will most certainly have a large strong polar cell. The polar cell has subsiding air at the poles and rising air at the polar front. On Earth about 60degrees.
With such a narrow landmass at the equator, you won't have a very strong zone of ascending warming air. Resulting in a weaker and smaller Hadley cell. This means that less warm air is going to be transported up into the upper atmosphere and down to the poles. Your tropics(where all the moisture falls back out if the sky) will also not reach to the 30 degree latitude and probably fall between 10-20 degrees (if that).
Your ferrel cell. This is where a very large amount of uncertainty lies. If it exists... Maybe just an interaction cell. Purely to exist to separate the cold polar air from the warmer equatorial air. It would be more of a buffer zone than an actual atmospheric cell.
I believe you have created a very cold planet. Your polar icecaps will dominant the weather. Most moisture will be locked in the salter than Earth's oceans and poles. The air will be dryer. It will rain alot on your equator landmass, as the rain band is that much narrower than on Earth. There will be strong polar winds. Storm surges onto your landmass probably won't be uncommon. Expect lots of coastal flooding in low-lying areas.