I have an old classic guitar. The bridge is made of very soft wood (or it got softer after a time, no difference). The problem is that the strings bend down in places they make contact with the bridge. 
So the strings get weaker after 1-2 minutes of playing and I have to set them up again. What to do?
P. S. I know that it is good to get anew bridge, but I have to play in few days and I'm looking for fast fix.
A couple of options for a fast fix:
Switch to ball end type nylon strings so you don't have to tie the strings over the top. You can also replicate this by tying a barrel knot at the end of the string and put the string directly through the hole with the knot holding the string in place (similar to how Ukulele strings are used).
If you don't mind making permanent modifications to the bridge, you could use CA glue (woodworking Super Glue) and fill along the edge and in the divots. The CA glue will be harder than the wood and reduce the indentation.
Looking at the picture, it looks like there is a channel or groove in the back edge of the bridge. It may be that this guitar is built to use the knot or ball method for the strings, which the slot would be for. In such a case, tying over the top isn't what the bridge is designed for, which is why the strings would be causing damage.
In some soft wood bridges I've seen fret wire installed on the edge, which isn't a quick fix, but could be an option to replacing the bridge. The more I look at the picture though, I think the bridge is designed to use knotted or ball end strings.
