Yes, Training a Naive Bayes Classifier for each category and then labeling each message to a class based on which Classifier provides the highest score is a standard first approach to problems like this. There are more sophisticated single class classifier algorithms which you could substitute in for Naive Bayes if you find performance inadequate, such as a Support Vector Machine ( Which I believe is available in NLTK via a Weka plug in, but not positive). Unless you can think of anything specific in this problem domain that would make Naieve Bayes especially unsuitable, its ofen the go-to "first try" for a lot of projects.
The other NLTK classifier I would consider trying would be MaxEnt as I believe it natively handles multiclass classification. (Though the multiple binary classifer approach is very standard and common as well). In any case the most important thing is to collect a very large corpus of properly tagged text messages.
If by "Text Messages" you are referring to actual cell phone text messages these tend to be very short and the language is very informal and varied, I think feature selection may end up being a larger factor in determining accuracy than classifier choice for you. For example, using a Stemmer or Lemmatizer that understands common abbreviations and idioms used, tagging part of speech or chunking , entity extraction, extracting probably relationships between terms may provide more bang than using more complex classifiers.
This paper talks about classifying Facebook status messages based on sentiment, which has some of the same issues, and may provide some insights into this. The links is to a google cache because I'm having problems w/ the original site:
http://docs.google.com/viewer?a=v&q=cache:_AeBYp6i1ooJ:nlp.stanford.edu/courses/cs224n/2010/reports/ssoriajr-kanej.pdf+maxent+classifier+multiple+classes&hl=en&gl=us&pid=bl&srcid=ADGEESi-eZHTZCQPo7AlcnaFdUws9nSN1P6X0BVmHjtlpKYGQnj7dtyHmXLSONa9Q9ziAQjliJnR8yD1Z-0WIpOjcmYbWO2zcB6z4RzkIhYI_Dfzx2WqU4jy2Le4wrEQv0yZp_QZyHQN&sig=AHIEtbQN4J_XciVhVI60oyrPb4164u681w&pli=1