No table saw or router? If you have a good skil saw you can
....usually find a "saw table" at a second hand store(usually 20-35 bucks) it's a table that your circular saw bolts to underneath to make a table saw-ish thing .. Be sure to make sure the rip fence is straight and true...actually check it with a full size framing square (16x24) or you are almost sure to regret it
....or make a jig that holds the saw and has a channel/guide under the saw to receive a specific size piece of wood and make a specific cut.
Both suck. saw table sucks less ... I looked on lowes site and amazon to give a "new" price and couldn't find one... maybe they've been outlawed because they suck and can be a bit dangerous if you don't bolt the table to something heavy.
If you don't mind long rip cuts or spending 1.50 to 2.00 per foot for the wood ... 1x8 lap siding is rabbeted on both sides. Actual measurements are usually 7.25" x 3/4" . be careful because the rabbets are different from each other .. 3/4" and 5/8" (both "widths" are 3/8" if you are holding it edges up , like sides of a box) ... Just rip cut off the undesired rabbet side of the board to leave desired wiidth ...to make a shallow box side ...
Or really down and dirty ... galvanized flat braces
lowes item #3627555
https://www.lowes.com/pd/Simpson-Strong-Tie-LTB-19-1-2-in-22-Gauge-Galvanized-Tension-Bridging/5001139065?cm_mmc=shp-_-c-_-prd-_-bdm-_-ggl-_-LIA_BDM_103_Roofing-Gutters-_-5001139065-_-local-_-0-_-0&gclid=Cj0KCQjw0tKiBhC6ARIsAAOXutnlI4xwPcnDFPMxhiLXcC29ycbt0dVna35o4iZWiLYq5w4hUBVEgyQaAioQEALw_wcB&gclsrc=aw.ds(
I actually used these to cover the frame rests on my boxes to make them more durable .. but you could just as easily make the non-rabbeted sides of the box 3/4" longer, reverse the braces to stick out like a little shelf It will scew up the "bee spacing" at the end of the frames without a 3/8" filler plate though ... and if you have something for a filler plate , you might as well rest frames on that ... to use these, cut the tab off one end where the 90 degree angle stops, cut to length, and bend the partial tab left on your piece to 90 degrees like the rest. drill 3 screw holes for mounting ... Warning .. the 90 degree bend is not actually 90 degrees .. If you are mounting it like a "shelf", doesnt matter: just mount a little lower. AS a cover to the frame rests, I re-bent the whole length because I had tools to do it, but you could just as easily clearance a rabbet cut with a chisel or block plane
Your "strip of wood" idea ... If I HAD to ..1/2" x 3/4" S4S moulding across the top to keep in the bees (box panels cut 3/4" shorter than other two and moulding attached hanging to outside as little a possible to still allow frame clearance ... a minimum of 5 hard attachment points (I would use 1.25" or longer 18ga. staples) and polyurethane glue along entire length... also would make it with the other sides of the box slightly taller (up to 1/32") than the "rest" sides.