You can install any Sinhala font from your mobile itself and use it in any browser you like with following instructions.
Preparation:
1) Root your device.
I rooted my Sony Xperia V (LT25i) with "towelroot" tool. No need to run complex scripts, just one click is enough.
2) Download a Sinhala ttf font to your phone and rename it to "DroidSansFallback.ttf" (or get a renamed file from here)
I have put this font file to "/sdcard1/DroidSansFallback.ttf" in the beginning.
3) Install a terminal emulator app from Google play store", I used this "terminal emulator"
Procedure:
Open the terminal emulator app and run the following commands in the given order.
1) Get root access
#su
2) Grant super user access to terminal emulator in the pop up window.
3) Get read/write access to /system
#mount -o rw,remount /system
4) Copy DroidSansFallback.ttf file /system/fonts
#cp /sdcard1/DroidSansFallback.ttf /system/fonts
Now you can open your favorite browser and read Sinhala text, I used Firefox browser to test it.