I purchased this exact same cajon kit. The instructions are really easy considering there are only a few pieces. Unless you put the front face on the bottom and left a whole side off, I am sure that you assembled it correctly. One thing that I did during construction was to counter-sink the screws holding the front face on. This was more for comfort as I did not like my hand slapping the screw heads.
As with many instruments, it takes some time to "get to know" your instrument.
That being said, if you look at the prices of what this same cajon costs already assembled from Meinl, you will see that it is not a top-of-the-line cajon. More expensive cajons may have adjustable snares, better woods or a different shape that may affect the acoustics.
For me, it has taken a bit of time, but I really like the sound I can get from my cajon. It just took me a while to figure out how hard (or soft) and where around the front face to hit to get the sounds I look for.