My grandfather remembers the war, which he was a part of. He is proud of the fact that they won, and he is also nostalgic as he remembers those days. He is/feeling _____________.
There is no single word that encapsulates both the emotions of nostalgia and pride. Perhaps your grandfather is feeling sentimental about the war he fought
(adj.) expressive of or appealing to sentiment, especially the tender emotions and feelings, as love, pity, or nostalgia
Or if you're looking for an idiom, consider
to be proud of someone or something
My nostalgic grandfather took a great deal of pride in winning the war which he was part of.