Can astronomers observe neutron stars optically? Are there any neutron stars near enough for astronomers to observe them optically?  If not, then how close are we to having the technology to do so?
 A: Yes, there are lots of optical observations of isolated (non-pulsar) neutron stars. Such observations have been done for many, many years.
An early example would be Kulkarni & van Kerwijk (1998) who detected optical radiation from the counterpart to RX0720.4-3125. The authors discuss various mechanisms that include thermal emission from the surface (if its young) or accretion from the ISM or magnetospheric processes.
Another example is a claimed thermal optical emission detection from the isolated neutron star RXJ185635-3754 by Walter & Matthews (1997). A picture of this is shown below, obtained by HST. 
Non-thermal emission detected from pulsars is even older news. Cocke et al. (1969) detected optical pulsations from the Crab pulsar. Several more pulsars, including millisecond pulsars have been detected at optical wavelengths (e.g. Shearer & Golden 2002; Sutaria et al. 2003).
Knowing the pulse period and phase of a known pulsar facilitates detection by allowing the accumulation of photons at particular times. Old-style photoelectric photometers were well suited to this.
A visible band HST image of the isolated neutron star RXJ185635-3754, attributable to Walter & Matthews 1997.

A: There have been a few detected in the visible range by Hubble Space telescope and Keck observatory. The magnitude is less than 25 for the Hubble image(PSR0656+14). Neutron stars are very hot, 600,000 or more Kelvin and very small (Hubble star was 28 kilometers in diameter) so their visible luminosity is very low. The emission is redshifted due to General Relativity effects but remains mostly in  the X-ray range for the black body peak radiation.
