Short answer: (really!)
Ocean: it depends on how big your continent is but normally, the ocean currents are influenced by 3 things: the winds, the Coriolis Effect and the landmasses
Winds are influenced by the difference of pressure between the
different regions.
The equator and its surroundings is typically a
low pressure zone. This is because it is the hottest place on Earth.
Hot air rises and the surrounding air masses converge there to fill
the gap. It's a convection movement. The air rise and then moves
toward the poles. It cool off with time and eventually the pressure
increase is dragging the air mass down near the 30th of latitude.
This is know as the Hadley cell.
On the poles, it is very cold. Here, the air is not ascending but
descending since it's a high pressure area. The air moves toward the
equator and is getting warmer until it reaches the 60th latitude. At
that latitude, the air movement is ascending. This is the Polar
cell.
It the middle, you have the mid latitude cell, or Ferrel cell. The
dynamic is mostly imposed by the other 2 cells and it just follows a
logical continuation of the same convection movement. At 60th of
latitude you have ascending air from the pole and from the mid
latitude. The air converge there.
At the 30th of latitude, you have descending air. The air in this
cell rise at 60th and sink back at 30th. If your a visual person like me, take a look at this if it's still
not clear: http://en.wikipedia.org/wiki/File:AtmosphCirc2.png
to summarize
Poles : sinking air= dry
latitude 60: rising air= humid
latitude 30: sinking air= dry
Equator: rising air=humid
*This is just a general rule. These areas are always moving on Earth
according to the seasons and are influenced by different factors
including the landmasses.
Rising air, this is when you have the precipitations. Other factors
also generate precipitations but this is the most important. Rising
air is hot near the land but cools off when ascending in the
atmosphere. The colder air cannot contain as much humidity as hot
air and this water needs to fall down. The sinking air is always dry
since it already got rid of most if not all the humidity it once
had.
Wind interaction with land: As I mentioned earlier, hot areas will generate low pressure. Your continent will always be hot since it sits on the equator. If it is large enough, it might trigger the mega monsoon mentioned by Ivy_lynx. It is not easy to extrapolate what we know to an extreme case. Asia is dragging the low pressure area of the equator up north as far as Shangdong China and even beyond. This is because the northern hemisphere gets really hot. I'm speculating a bit here, but in your case the continent (unless it is stretching far from the equator), would probably not have this dragging effect. The low pressure zone would move north and south but not enough to meet the low pressure of the 60th latitude.
2- Coriolis effect: it tends to deflect the currents in a clockwise manner in the northern hemisphere and counter clockwise in the southern hemisphere. By itself, it doesn't make the currents, it really just deflects them. http://en.wikipedia.org/wiki/Coriolis_effect
Coriolis+wind= it's the same result as with water. At the equator and at the mid latitudes, you have a low pressure area that is drawing the air in. Near the equator, the Coriolis effect is directing the winds and water currents toward the west but towards the east in the mid latitudes.
3-Interaction: water+land+Coriolis: the water currents will converge at the equator. At this latitude, the water flows east to west. Then, when it reaches the coasts of the continent, the water will be deflected toward each poles.
The next step depends on the size of the continent. Normally, it would flow towards the pole until the mid latitudes between 30th and 60th and then be pushed toward the east by the dominant winds. One part of the water will probably go poleward of the continent following the dominant winds (west to east). Then, my guess is that without any other landmass, most of the water will reach the opposite shore of the continent and will be deflected back toward the equator.
At the poles, water would just spin around like it does around Antarctica.
I made you a very nice image :http://sketchtoy.com/63277181
I think I've covered all the bases. I just need to know the specific information to give a more precise answer. But I might not be able to
