The adapter bracket probably will work. In fact, check this bit carefully before ordering one. Many external HDD units with eSATA ports come with an adapter plate exactly like we're talking about, so you don't have to buy separately. Since it appears you have not ordered and received the drive yet, ask the supplier whether it will come with the adapter already.
My "probably" comment is based on this. The adapter very simply converts one connector form (SATA) into another (eSATA). The electrical connections for the two port types are the same, so this works just fine. What is not quite guaranteed is a few extra features of eSATA compared to SATA. The two that come to mind prominently are hot swapping and long cables. "Hot Swapping" in true eSATA systems means that you can disconnect or connect the eSATA device to your computer at almost any time, even with power on and Windows running, without causing any trouble. "Long cables" just means that true eSATA uses slightly higher voltages on the signal lines so that the maximum length of the 7-conductor ribbon cable for data can be twice as long and still get clean error-free signals through. Neither of these (and a few others) was part of the official SATA design. However, many makers of regular SATA chips and ports on mobos today also include several of these eSATA features. Hence, using an adapter like this does truly give you an eSATA port, because of those added features in the "plain" SATA ports of the mobo. The trick is, you don't really know whether those added features are there or not.
What to do? Go ahead an use the adapter. If you want to be safe, do not try to do Hot Swapping. Just shut your system down before disconnecting or connecting the eSATA unit from the computer. As far as cable length goes, if it works smoothly and you don't get errors or slow HDD performance caused by repeated access attempts, then it's working and you should not worry.
eSATA is significantly faster than USB2. In my case, my mobo actually does have a true eSATA port; my eSATA unit's speed is the same as my internal SATA drives.