Ubuntu actually has an IME for Sinhala, you just need to add it via system settings -> keyboard layout.
There is also a translation team on launchpad that would be glad to welcome new members and could give you better advice on related issues.
On that wine issue
You got an error message from a windows application by using wine. Windows applications are generally unaware that they are running on wine and will assume they are running on windows and like in that case will make false assumptions for workarounds and solutions.
The default configuration of Ubuntu has two firewalls. The kernel firewall named Netfilter, which is not configured to filter anything by default as far as I know, and Apparmor an application based firewall which shouldn't get in the way of wine applications.
What zetah referred to was the appdb on the WineHQ site, where a compatibility list of applications is maintained. Unfortunately you can't find "Google Sinhalese" there or variations there of. But what you can find is a reference to a similar error message of another application using the Google Updater. Searching for other Google applications, Google Music Manager seems to work, but trying to reproduce that with the Sinhalese IME installer in my VM did not work. Filehippo has offline installers for Chrome, but does not seem to have for the IME.
Sorry but without more investigation, you would not be able to install that application. Note: We are just talking about a file that needs to be fetched from a server, that's all there is, the rest of the installation process would run fine. It is a nuisance, but Google still insists on using this malware like installation process for their Windows software. They have been doing so or years.