“theruwan saranai” and "budu saranai" are used exclusively by Sinhalese Buddhist to wish/greet other Sinhalese Buddhist. Its not supposed to be use elsewhere because they are religious greetings.
Theruwan – තෙරුවන් - mean the triple gem in Buddhism. i.e : Lord Buddha , his dhamma (Buddhism) and his disciples (Sangha/monks).
Budu - බුදු - is a shorter form for Lord Buddha
Saranai – සරණයි - means taking/going refuge of something . But in this context it means that may you be protected/helped/supported by xyz.
So all in all it means
- theruwan saranai - තෙරුවන් සරණයි - May you be protected by triple gem
- budu saranai - බුදු සරණයි - May you be protected by the noble power of Lord Buddha
Ganbatte kudasai ^_^