"Is it necessary to initialize/reset the RAM bytes to have a value of 00xh starting before the codes starts??"

No, it is not necessary; however, it is essential that you ensure that your variable contains a valid value before you use it!

Utilities like lint, and many compilers, will warn you of "uninitialised" variables.

"Routine_1 comes first in the code where the byte has to be zero or the result would corrupt."

In that case, you the programmer must ensure that it is zero - whether by specifically initialising it in your code, or by having a general start-up process which clears all your RAM.

"I never got rid of the problem untill I made the byte zero before the Routine_1."

See above: if a routine requires a certain value to work correctly, you must make sure that it gets that value - otherwise, by definition, it won't work, will it?!

"I was assuming that the RAM bytes are supposed to contain zeros at the power up."

A totally unwarranted assumption!
If you're a 'C' programmer, this may appear to be the case - but that's only because the 'C' runtime support does the initialisation for you!