Note: As was discussed in the comments, this is an unusual situation. Usually you would either have depth measurements as points, or a DEM raster. Starting with depth measurements as points, you would interpolate them into a DEM, then use the DEM to generate contours.
In this case, the only data available is contours without depth measurements, and a pdf map with depth labels. For anyone who finds themselves in this situation, I strongly recommend searching for better data. Use the method below only after you've exhausted the possibility of finding proper data.
Here's a semi-automated method to apply known depth values to contours that came without depth values.
Create a new line layer. It can be a temporary scratch layer. Draw a line that crosses from the outside (shoreline) contour to the deepest contour. (The direction of the line matters.) Choose a location where the cross line will increase in depth from start to finish, IE not crossing any areas where the lake get shallower.
Line Intersection tool to get the points where the cross line crosses the contours.
Assign depth values to the intersection points with a calculation like:
(-1 * (@row_number -1)) * conversionfactor
The conversion factor is the spacing between the contours. To figure this out, count how many contour intervals there are from the shoreline to the deepest point (without crossing any rings). Divide this number by the depth of the lake. EG, my lake is 4m deep, and there are 4 intervals, so the contours are 1m apart, and my conversion factor is 1.
(-1 * (@row_number -1)) * 1
Join Attributes by Location tool to copy the depth values from the points to the contours.
Note: Joining by location failed to join any of the features. This might be a bug, but apparently the intersection points don't actually intersect with the contours (they should, but they don't). As a workaround, I used the NNJoin plugin to do a nearest-neighbor join.
This creates a new layer of contours with depth values:
Note: If your contours are discontinuous, you'll need to repeat the process a few times until all the contours have depth values. Manually label contours in areas of shallow water surrounded by deeper water.