Add color to boxplot - "Continuous value supplied to discrete scale" error

First, providing sample data would help. Since you didn't, here is some:

MYdata <- data.frame(Age = rep(c(0,1,3,6,9,12), each=20),
                    Richness = rnorm(120, 10000, 2500))

Parts 1 and 2 stem from the same problem. Age is a continuous variable, but you are trying to use it in a discrete scale (by specifying the color for specific values of age). In general, a scale maps the variable to the visual; for a continuous age, there is a corresponding color for every possible value of age, not just the ones that happen to appear in your data. However, you can simultaneously treat age as a categorical variable (factor) for some of the aesthetics. For the third part of your question, within the scale description, you can define specific labels corresponding to specific breaks in the scale. Putting this all together (and adding something to give you the x axis labelled more like what you have in the example):

ggplot(data = MYdata, aes(x = Age, y = Richness)) + 
  geom_boxplot(aes(fill=factor(Age))) + 
  geom_point(aes(color = factor(Age))) +
  scale_x_continuous(breaks = c(0, 1, 3, 6, 9, 12)) +
  scale_colour_manual(breaks = c("0", "1", "3", "6", "9", "12"),
                      labels = c("0 month", "1 month", "3 months",
                                 "6 months", "9 months", "12 months"),
                      values = c("#E69F00", "#56B4E9", "#009E73", 
                                 "#F0E442", "#0072B2", "#D55E00")) +
  scale_fill_manual(breaks = c("0", "1", "3", "6", "9", "12"),
                      labels = c("0 month", "1 month", "3 months",
                                 "6 months", "9 months", "12 months"),
                      values = c("#E69F00", "#56B4E9", "#009E73", 
                                 "#F0E442", "#0072B2", "#D55E00"))

With this color scheme, the points that fall inside the boxplot are not visible (since they are the same color as the boxplot's fill). Perhaps leaving the boxplot hollow and drawing its lines in the color would be better.

ggplot(data = MYdata, aes(x = Age, y = Richness)) + 
  geom_boxplot(aes(colour=factor(Age)), fill=NA) + 
  geom_point(aes(color = factor(Age))) +
  scale_x_continuous(breaks = c(0, 1, 3, 6, 9, 12)) +
  scale_colour_manual(breaks = c("0", "1", "3", "6", "9", "12"),
                      labels = c("0 month", "1 month", "3 months",
                                 "6 months", "9 months", "12 months"),
                      values = c("#E69F00", "#56B4E9", "#009E73", 
                                 "#F0E442", "#0072B2", "#D55E00"))

Finally, consider if you really need to color each age differently, since they are well defined by the x-axis already.