I would not use facets in your case, because there are only 3 variables.
So, to get a similar plot in R using ggplot2, you first need to reformat the dataframe using gather() from the tidyverse package. Then it's in long or tidy format.
To my knowledge, there is no geom that does what you want in standard ggplot2, so some fiddling is necessary.
However, it's possible to produce the plot using geom_segment() and cumsum():
library(tidyverse)
# First reformat and calculate cummulative sums by type.
# This works because factor names begins with A,B,C
# and are thus ordered correctly.
df <- df %>%
gather(-type, key = "route", value = "time") %>%
group_by(type) %>%
mutate(cummulative_time = cumsum(time))
segment_length <- 0.2
df %>%
mutate(route = fct_rev(route)) %>%
ggplot(aes(color = route)) +
geom_segment(aes(x = as.numeric(type) + segment_length, xend = as.numeric(type) - segment_length, y = cummulative_time, yend = cummulative_time)) +
scale_x_discrete(limits=c("1","2","3"), labels=c("Z", "Y","X"))+
coord_flip() +
ylim(0,max(df$cummulative_time)) +
labs(x = "type")

EDIT
This solutions works because it assigns values to X,Y,Z in scale_x_discrete. Be careful to assign the correct labels! Also compare this answer.