I guess without sudo you cannot install google chrome. Because google chrome will try to install it in directory /opt and few other root directory, which needs sudo privilege.
But the alternative way is to install the chrome portable app. But to run a portable app you must have RUNZ framework
Follow the procedures in this link to install google chrome portable.
But I would recommend you to install the Google chrome with sudo permissions using the below commands
uGet is the best Linux Download Manager available and works great on Ubuntu 12.04
We (uGet team) are also in the process of developing a Google Chrome/Chromium Extension that does exactly what you are requesting. The extension is still in development but we hope to release it as soon as possible.
Note* - Ubuntu Software Center has a VERY OLD version of uGet regardless the amount of effort we have put in to get it updated. So for the latest version go to http://ugetdm.com/downloads
Best Answer
According to the Google Chrome website, Google has not made Chrome available for Ubuntu. Please contact them for further details.
https://www.google.com/chrome/browser/mobile/index.html