Structure of NN for input data with drop out
There is no Special reason that neural networks are suitable, if anything it could be too much (depending on the complexities of your time-series).
In any case, you need to deal with you Nan values and if you are looking for a fast model, that imputes that automatically, I would suggest lgbm. Since it will impute NAN with values that reduce the out of fold loss the most (making the highly optimal imputation).