Assuming your OBi connects to the internet and gets its local IP settings by DHCP or manually and there is no blocking carried out by the ISP, then yes it should work. In particular I would expect Japan to work - I have an Obi100 there which uses the OBitalk network to communicate with the UK. South Korea will probably work. North Korea and China?
One potential issue is internet access in hotels. Some assume the connected device will be a computer with a browser and present a log-on page to be completed before access is granted. An Obi device cannot handle that. And some hotels do not provide a wired ethernet socket, only Wi-Fi.
And the connection needs to give you a sustained data rate of more than about 100 kbit/s.