Can you reverse the arrow of time? Has there been any attempt to reverse the arrow of time? 
 A: First, it should be noted that "reversing the arrow of time" is not the same thing as "traveling back in time" (of course you didn't say this, but just in case other readers have this idea in their mind). Second, you are probably more familiar with the reversal of time's arrow than you realize.
The arrow of time is just a different way to view the second law of thermodynamics. Really, the arrow of time is just our perception of entropy increasing. For example, if we were able to see all of the gas molecules in a room suddenly gather in the corner of the room without any external forces acting on the particles, it would look like the gas molecules were "moving backwards in time", since we would normally expect gas molecules over time to diffuse away from being highly concentrated in the corner of a room.
In contrast, when entropy is at a maximum (or not definable) the arrow of time loses any direction or meaning. For example, let's take our same gas in our room at equilibrium (maximum entropy) and record a video of the gas for a long time. Now, let's run the tape in reverse. It would look just like if we were playing it forward. You would not be able to distinguish between backwards and forwards. The lack of entropy increase has removed the perception of time flowing in some direction.
Therefore, to answer your question, most certainly. We do it a lot. Just look for a system whose entropy is decreasing. For example, the food in your refrigerator. If you were to only focus on some warm food you just put into your refrigerator, you would see energy leaving the food as the motion of the food particles became less random; energy would flow from being "less concentrated" to "more concentrated". At this level, you could conclude that your refrigerator has reversed the arrow of time.
Of course, as we know, the decrease in entropy from the refrigerator will be accompanied by an increase in entropy outside of the refrigerator, but a decrease in entropy (reversal of the arrow of time) has occurred somewhere. Therefore, the arrow of time was reversed there as well.
All you need to do is decrease the entropy of a system, and you will have reserved the arrow of time for that system.
Of course, if you mean to ask if we can universally reverse the arrow of time, then the answer is no. As you can probably tell, this would mean reducing the entropy of the universe. Unless our universe is not a closed system and we can find a way to have some external "forces" that can decrease the entropy of the universe, then this is not possible. Typically we assume the universe is a closed system, so entropy must always increase (until the heat death of the universe), and therefore the arrow of time will always be "forward".
A: Actually, yes. And the arrow of time has been experimentally reversed in a chloroform molecule in a NMR setup.
The "trick" here is that in normal discussions about thermodynamics different systems are assumed to be uncorrelated: thermal motions are random and part A and part B of the system will not give any information about the state of the other part. If there is mutual information one can make (microscopic) systems that have heat flowing from cold to warm parts or otherwise break the arrow of time. The decrease in mutual information compensates for the entropy reduction (and this gets potentially bigger for quantum mutual information).
On a macroscopic scale this is unlikely to work very well, since there are too many degrees of freedom to manipulate.
