Do we know a star that is similar to the Sun when it would be a red giant? In about 5 billion years the Sun is predicted to become a red giant and have more than 200 times its current size, reaching a radius of about 5 AU when largest. 

I wonder what spectral class the Sun would have then, I think somewhere from M1III to M5III. Do we know a red giant star that is similar to what the Sun is predicted to become?
Models for the future behaviour of the Sun do vary, mainly as a result of uncertainty of mass loss during the red giant (H shell burning) and asymptotic red giant (H+He shell burning) phases.

A highly cited paper by Schroeder &amp; Smith 2008 claims that the Sun will reach its maximum size of about $256 R_{\odot}$ (1.18 au) at the very tip of the red giant branch (and not at the end of the asymptotic giant branch phase, which is suggested by some other models). This maximum size will occur 7.6 billion years in the future (not the 5 billion years of popular literature), when the Sun's mass will be reduced to about $0.7M_{\odot}$ and have a surface temperature of 2600 K (or about 2300 Celsius). This would have a spectral type of M6III, or perhaps even M6I.

Do we know of a star similar to this? It depends what you mean by similar, but there are unlikely to be any stars like this in our Galaxy. The reason for this is that star formation began in our Galaxy about 12 billion years ago. But a 1 solar mass star like the Sun requires about 12 billion years (or a bit more) to reach the tip of the red giant branch. Even were this just about possible time-wise and we were to find a red (super)giant star of about $0.7 M_{\odot}$, it would be highly unlikely that the star would have a similar chemical composition to the Sun. That is because stars born early in the life of our Galaxy would have very low concentrations of metals like iron and nickel that are only produced inside stars and only present in the interstellar medium when a generation of stars have lived and died.

So my perhaps pedantic answer, is that there aren't any big red giants we can see now that started life as 1 solar-mass stars and have a similar chemical composition to the Sun.