Figtree should recognize that your tree has labeled branches - when you open a tree from a file it will ask you to assign a name to the labels. Say you name it "my_labels", you can view them by expanding the "Branch labels" (or "Node labels") bar in the options panel at the left, then click "Display" and select "my_labels" and they will show up.
Another easy viewing option is in R. For manual labeling and checking, I find this faster than Figtree. Install the 'ape' package, then:
my_tree<-read.tree("path/to/newick/file") # read in the file, or alternately below:
my_tree<-read.tree(text="((your, raw)#1, (labeled, newick));") # read in the raw newick text
my_tree<-ladderize(my_tree) # for easier viewing, optional
plot(my_tree, show.node.label = T, cex = 0.5) # cex is optional, but you may want to try smaller numbers to help with viewing if your tree is large
Just check that your labels appear in the right place and if not, manually edit the newick text and repeat above to check again.
Node labels can also be coded out in ape if that is more useful for you.
Info here.
Hope that helps!
-Ben