What to do for missing points in CPI time series?
Statistical Analysis with Missing Data by Little and Rubin is the go-to reference for working with missing data, at least if nothing state of the art is required. In general, this is a complex problem that remains an area of active research. The relatively easy cases are when the data is missing-completely-at-random or missing-at-random. Even among the most basic single imputation methods you have a lot of choices (list from Little and Rubin):


Mean imputation (replace with mean values)
Regression imputation
Stochastic regression imputation
Hot deck imputation (substitute individual values drawn from "similar" response units)
Substitution (not relevant in your context)
Cold deck imputation (replace missing value with constant value from external source like last value)
Composites of the above methods 


However, if all the the general price level is what's of interest and not the dynamics of the price level (e.g. because you want a deflator instead of studying inflation dynamics) linear interpolation / extrapolation may be just fine. Fundamentally, since deflation is rare, if prices are 100 at time t and 110 at t+2, realistically prices at t+1 are going to be somewhere in $[100,110]$ and lots of models can get you there.  

You can check the within and out of sample prediction quality to asses if your method is a good predictor of the missing prices. Within sample testing could be as simple as asking if the $R^2$ is high of predictive model. Obviously you can do much richer analysis than that. For out of sample testing, consider splitting the sample and calibrating the model only on the first half of the data, then evaluating prediction quality on the second half of the data.