Moving company
Each moving company is different. They have different strengths and weaknesses; they employ different people; they own different equipment; most importantly, they have different sizes of moving vehicles. Generally, the larger a moving vehicle is, the more it will cost to hire it for your move. If you are only moving a small apartment, paying for a large moving truck is a waste of money. Before you book a moving company, do some research. Although this is always a good idea, what you are looking for specifically are the vehicles that they own. You don't want to hire a moving company that only has large moving trucks available.
Packing
When moving out of a large home, it is important that packing be started well in advance. When moving out of a small apartment, however, this is not the case. The rule of thumb for a one-bedroom apartment is that it takes around 1-2 days to pack. If you clear your schedule, it should be very doable over a single weekend. That said, if you are unable to commit entire days, you may have to start earlier and do it over time. You should start by packing the things that you don't need; Leave anything that is used daily until the end.
You want to make sure not to pack things too early. As soon as something is packed, it becomes a lot more difficult to access. You don't want to waste your time going through boxes every time you need something. This is especially true for items such as plates and glasses as these items will need to be unwrapped and then rewrapped whenever you use them. With how little time it takes to pack a small apartment, you should have no reason to pack anything too early.