For the dependent variable, the usual starting point is ordinal logistic regression. This makes some assumptions (chiefly that of proportional odds) and there are ways of dealing with violations of this. These have been discussed here before.
For the independent variable, things are a little murkier. Some people treat it as categorical, some as continuous (e.g. by coded them with numbers). One general approach is optimal scaling. I don't know if this is available in
For your particular variable, one approach that I have used is to convert it into times per month. So, daily - 30, weekly = 4, monthly = 1, never = 0. You can do a sensitivity analysis by changing these a bit (after all, people who answer "daily" might not be exactly 30 times per month).