Yes, you may use rolling correlations and standard deviations to get rolling beta estimates.
The time horizon used is your investment decision horizon: in other words, how often might you adjust your portfolio? If you revisit your portfolio's investments every year, then the time horizon should be a year. This also affects the risk-free rate you use when calculating excess returns for the stock and market index.
Daily data is not necessarily the best for estimating betas. Daily data is slightly polluted by bid-ask bounce, so that added source of noise will tend to bias beta estimates to be lower than estimates produced using weekly, monthly, or longer-term data.
The tougher question is how much data you should use for estimation: one year back? three years? more? The answer gets very complicated (see the statistical work of Allan Timmerman on optimal windows for mean vs variance estimation). Barring tackling that question with a complicated method, you can do as many people do and use three years prior.
Finally, I would be remiss if I did not point out that the CAPM betas will vary across the business cycle and suggest the presence of alpha more often than we would expect from economic theory. This is because your model is too simple: you should use a multi-factor model. Typically, adding in other factors like small vs large firms (e.g. outperformance of Russell 2000 vs the S&P 500), changes in the level and slope of the yield curve, momentum, inflation, and a credit spread (e.g. IBoxx IG - UST 10Y) will yield coefficients that are more stable across the business cycle; and, the alpha will be much closer to 0.