How do I select the first row in an R data frame that meets certain criteria?

Sure. Something like this should do the trick:

# CREATE A REPRODUCIBLE EXAMPLE!
df <- data.frame(year = c("2001", "2003", "2001", "2003", "2003"),
                 pixel = c("a", "b", "a", "b", "a"), 
                 cumsum = c(99, 99, 98, 99, 99),
                 numbercomps=1:5)
df
#   year pixel cumsum numbercomps
# 1 2001     a     99           1
# 2 2003     b     99           2 
# 3 2001     a     98           3
# 4 2003     b     99           4
# 5 2003     a     99           5

# EXTRACT THE SUBSET YOU'D LIKE.
res <- subset(df, cumsum>=99)
res <- subset(res, 
              subset = !duplicated(res[c("year", "pixel")]),
              select = c("pixel", "year", "numbercomps"))
#   pixel year numbercomps
# 1     a 2001           1
# 2     b 2003           2
# 5     a 2003           5

EDIT Also, for those interested in data.table, there is this:

library(data.table)
dt <- data.table(df, key="pixel, year")    
dt[cumsum>=99, .SD[1], by=key(dt)]