I managed to find a post on Physics StackExchange :
https://physics.stackexchange.com/questions/179393/any-tips-on-evaluating-riemann-tensor
which details what I feared. The process it self is too tedious. The best solution is to use a symbolic package which can do all these calculations, like for example the python package SymPy.
Another answer (from the same thread) is based on some thing called "Cartan formalism" (it is also the answer which has been up-voted the most).
