To get Cronet tests to run on a JB device I had to put in the following changes:

--- a/build/android/pylib/valgrind_tools.py
+++ b/build/android/pylib/valgrind_tools.py
@@ -20,7 +20,8 @@ def SetChromeTimeoutScale(device, scale):
   path = '/data/local/tmp/chrome_timeout_scale'
   if not scale or scale == 1.0:
     # Delete if scale is None/0.0/1.0 since the default timeout scale is 1.0
-    device.RemovePath(path, force=True, as_root=True)
+    if device.PathExists(path):
+      device.RemovePath(path, force=True, as_root=True)
   else:
     device.WriteFile(path, '%f' % scale, as_root=True)

And in third_party/catapult:

--- a/devil/devil/android/device_utils.py
+++ b/devil/devil/android/device_utils.py
@@ -2485,7 +2485,9 @@ class DeviceUtils(object):
     """
     process_name = process_name or ''
     processes = []
-    for line in self._GetPsOutput(process_name):
+    for line in self._GetPsOutput(None):
+      if process_name not in line:
+        continue
       row = line.split()
       try:
         row = {k: row[i] for k, i in _PS_COLUMNS.iteritems()}
--- a/devil/devil/android/md5sum.py
+++ b/devil/devil/android/md5sum.py
@@ -98,8 +98,9 @@ def CalculateDeviceMd5Sums(paths, device):
       # If files were previously pushed as root (adbd running as root), trying
       # to re-push as non-root causes the push command to report success, but
       # actually fail. So, wipe the directory first.
-      device.RunShellCommand(['rm', '-rf', MD5SUM_DEVICE_LIB_PATH],
-                             as_root=True, check_return=True)
+      if device.PathExists(MD5SUM_DEVICE_LIB_PATH):
+        device.RunShellCommand(['rm', '-rf', MD5SUM_DEVICE_LIB_PATH],
+                               as_root=True, check_return=True)
       if os.path.isdir(md5sum_dist_path):
         device.adb.Push(md5sum_dist_path, MD5SUM_DEVICE_LIB_PATH)
       else:

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/d79719a90b915d7fd60fe334cb34d9149dd8c464

commit d79719a90b915d7fd60fe334cb34d9149dd8c464
Author: Paul Jensen <pauljensen@chromium.org>
Date: Fri Jan 18 15:26:46 2019

[Cronet] Fix CronetUrlRequestTest.testManyRequests to run on Jelly Bean

Jelly Bean has a 2000 Java global reference limit.  Adjust test to
create slightly fewer requests so it stays under the limit on
Jelly Bean.

Bug: 922656
Change-Id: Idf42e04b1e3f9d716d17f3d6293831b650f19da5
Reviewed-on: https://chromium-review.googlesource.com/c/1421777
Auto-Submit: Paul Jensen <pauljensen@chromium.org>
Reviewed-by: Misha Efimov <mef@chromium.org>
Commit-Queue: Paul Jensen <pauljensen@chromium.org>
Cr-Commit-Position: refs/heads/master@{#624130}
[modify] https://crrev.com/d79719a90b915d7fd60fe334cb34d9149dd8c464/components/cronet/android/test/javatests/src/org/chromium/net/CronetUrlRequestTest.java

I spent some time trying to get the socket tagging tests to pass when run on JB.
It appears socket tagging byte accounting in /proc/net/xt_qtaguid/stats isn't done for loopback interfaces like it is on KitKat and newer devices.
I spent quite a while getting a test HTTP server bound to WiFi's address but this didn't get the bytes accounted for sadly :(
Here's my attempt: https://chromium-review.googlesource.com/c/chromium/src/+/1421682

The simple way to get these tests passing on JB is to change NativeTestServer.getEchoMethodURL() to "http://example.com" and run the test on a device with an Internet connection.