Is it possible to train the sentiment classification model with the labeled data and then use it to predict sentiment on data that is not labeled?
Is it possible to train the sentiment classification model with the labeled data and then use it to predict sentiment on data that is not labeled?

Yes. This is basically the definition of what supervised learning is.
I.e. you train on data that has labels, so that you can then put it into production on categorizing your data that does not have labels.
(Any book on supervised learning will have code examples.)
I wonder if your question might really be: can I use supervised learning to make a model, assign labels to another 500 articles, then do further machine learning on all 600 articles? Well the answer is still yes, but the quality will fall somewhere between these two extremes:

Assign random labels to the 500. Bad results.
Get a domain expert assign correct labels to those 500. Good results.

Your model could fall anywhere between those two extremes. It is useful to know where it is, so know if it is worth using the data. You can get an estimate of that by taking a sample, say 25 records, and have them also assigned by a domain expert. If all 25 match, there is a reasonable chance your other 475 records also have been given good labels. If  e.g. only 10 of the 25 match, the model is much closer to the random end of the spectrum, and using the other 475 records is probably a bad idea.
("10", "25", etc. are arbitrary examples; choose based on the number of different labels, and your desired confidence in the results.)
I want to do sentiment analysis using machine learning (text classification) approach. For example nltk Naive Bayes Classifier.
But the issue is that a small amount of my data is labeled. (For example, 100 articles are labeled positive or negative) and 500 articles are not labeled.
I was thinking that I train the classifier with labeled data and then try to predict sentiments of unlabeled data. 
Is it possible? 
I am a beginner in machine learning and don't know much about it. 
I am using python 3.7. 
Thank you in advance.