No. The fat content of goose, like most waterfowl, is very high. This makes it unsuitable for brining. Brining is intended to bring moisture to meats that tend to dry out, such as chicken and turkey.
See my answer to "What are the basics of brining meat?" for more details.
Meat is tough for two reasons:
1- An abundance of connective tissue.
2- When over cooked.
In your case I'd say you probably have both problems. Cheap meat is tough meat. It is from older animals or well worked muscle groups. This means that it has been fortified with a lot of extra connective tissue. It also means it has a lot of flavor.
The solution to #1 is slow, wet cooking that will melt that connective tissue into delicious gelatin. Braising is the normal way to do this. When meat is heated too far, even if the connective tissue has been carefully melted out, the meat proteins bunch up and stiffen- resulting in #2, a dry, unpleasant meal.
You are buying tough meat and cooking it relatively quickly with no thermometer. You don't have enough time or moisture to melt the collagen and you can't be sure you haven't over cooked the meat because you don't know the temperature. Using time doesn't work because chunks of meat are irregularly shaped so you can't know how long it will take for the heat to penetrate.
Buy yourself a probe thermometer to prevent #2. For #1 look for pot roast recipes. Some are easily done in slow cookers- others use a tent of sealed foil over the meat to seal in moisture. Plan on it taking much longer than your 1.5 hours. 3-6 hours are typical to produce a really succulent pot roast.
The searing is just for flavor and will not play a role in either melting collagen or cooking the interior of the meat.
Best Answer
As @ElendilTheTall says, very correctly, cooking to time is a bad idea. However, it's good to have a working time figure for planning. In the case of goose the timeline is to cook 20-25 minutes per pound. 4.6 lb is a very small goose so I'm assuming it's 4.6kg, which is just over 10lb, so half that is 5 lb, which should take somewhere close to between 1 hour 40 minutes to 2 hours to cook. You would need to add at least 30 minutes for resting after cooking as well.
I would aim to have the goose in the oven about 2:45 before you want to serve, and then keep a close eye on the goose's temperature. Use an oven thermometer that you can leave in the bird in the oven, it's much easier than having to pull the goose out and check it every 5 minutes. A handheld one is also useful to back up your measurements in case your leave-in probe is out of position. Do not overcook goose, keep a good eye on that temperature because goose can go from underdone to overdone in a very short time.