Well, if you are all using the same network, he can probably sniff your packets, which means that he/she can see all the traffic coming from and going towards your computer (your wireless card to be more precise). This is pretty much unavoidable.
What you can do, is to visit https version of sites, since then he/she can only see what site are you visiting, but not what data are you transmitting (password, chats, etc.). Generally speaking, you should always use https to connect to sites.
Also, if you want to be on the safe side, you could always pay for some personal VPN, which will encrypt ALL traffic to and from your computer, and they are fairly cheap. You can find many under 10$ a month, and they do provide very good encryption.
However, if you are concerned about traffic usage, there is no way to hide it, because even if he cannot "see" what you are doing, he can measure it. That means if you download 20 gigabytes of data, he will sure be able to measure how much did you download.
Measuring the downloaded amount is something that does not require advanced knowledge to get to work. Most routers these days can display traffic amount for each ip address associated to it, directly in their configuration/status panel, which the owner of the router probably has access to.
On the other hand, sniffing the data and getting your passwords, chats. etc does require that you visit non-encrypted site, but also requires some level of technical knowledge. Although with youtube tutorials around, I wouldn't put it past anyone. That is why visiting https version of sites is highly recommended (if not a must). Also, personal vpn is good investment if you are using untrusted hotspots (public for example, or even the one you are renting now).