To get citations in author-year format with the natbib package, be sure to load it with the option
authoryear, as in
The formatting of the entries in the bibliography itself is not governed by
natbib directly but by the bibliography style file you use. I'm not sure if I understand one of your remarks correctly, but you seem to indicate that you're using the
unsrt bibliography style. As its name suggests, it does not sort the entries in any way, i.e., they're listed in order of the corresponding citation commands. If you want to have the entries in the references section sorted alphabetically, you should use a bibliography style such as plain.bst or plainnat.bst. (Both of these files are included automatically in any reasonably modern TeX distribution.)
There are literally hundreds, if not thousands, of BibTeX bibliography style files in existence and available for downloading from either the CTAN or other repositories. Without knowledge of the precise formatting requirements you need to satisfy, I'm afraid it's not possible to give detailed advice on the style file you should choose.
Addendum, posted after the OP added some more information. First, to get a bibliography where the entries are sorted alphabetically by the author's (or authors') surnames, choose any of the many bibliography style files that performs alphabetical sorting; leading candidates for this assignment would be, as noted above,
plain.bst or its younger relative,
plainnat.bst. You mention that there are fairly strict guidelines for the formatting of your thesis, including the formatting of the bibliography. Please consider posting these guidelines so that people might be in a position to give advice on whether any ready-made bibliography style files could be used.
Second, to the best of my knowledge, there are no citation management packages that sort the arguments of a citation command chronologically -- especially if you use an authoryear citation system and the authors can have different surnames. (For numerical citation styles, there is the
cite package that sorts and compresses citation numbers, but that wouldn't appear to be of relevance for your case.)