Does the training set of one topic will be useful to predicate the sentiment for any other topic? I am working on Twitter sentiment analysis. Now, I should prepare a training data set that fits for any kind of Twitter data that predicts the sentiment of a tweet as pos, neg, and neu.

From googling, I found an air-line tweets training data set with needed labels (pos, neg, neu, and sentiment). When I checked the sentiment of tweets with that training data set, it gave me mixed results. I found that some positive tweets are given as negative and negative as positive. 

Is the method that I am following correct or not?

Please suggest your views for preparing a training data set that can predict any kind of the twitter data.
Since you have trained your model on air-line tweets, the model will learn the characteristics of the air-line tweets. There could be words used in air-line tweets which are positive and contain great weightage but never even used in other tweets leading to negative results. 
I suggest you to try 


Get a dataset which contains all kinds of tweets which you desire essentially similar to what your task ahead is.
Look into Transfer learning.