ICU-20482 Add Appveyor CI Build config changes

(Using the file @ 2e86b08fcd).

.appveyor.yml

@@ -1,8 +1,68 @@
-image:         Visual Studio 2017
-platform:      x64
-configuration: Release
-build:
-    project: icu4c/source/allinone/allinone.sln
+# AppVeyor configuration for CI Builds of ICU4C.
 
-test_script:
-    - icu4c/source/allinone/icucheck.bat x64 Release
+image: Visual Studio 2017
+platform: x64
+
+# Don't clone the entire repo.
+clone_depth: 3
+
+# Cache things between builds to speed things up and save bandwidth.
+cache:
+  - c:\icu-ci-cache
+
+environment:
+  global:
+    ICU_CI_CACHE: c:\icu-ci-cache
+    CYG_URL: https://cygwin.com/setup-x86_64.exe
+    CYG_MIRROR: http://cygwin.mirror.constant.com
+    CYG_PACKAGES: automake,gcc-core,gcc-g++,make,pkg-config,perl,python3
+    CYG_ROOT: c:\cygwin-root
+    CYG_CACHE: '%ICU_CI_CACHE%\cygwin64'
+    CYG_CACHED_SETUP: '%CYG_CACHE%\setup.exe'
+
+  matrix:
+    - BUILDSYSTEM: VS2017
+      configuration: Release
+      APPVEYOR_CACHE_SKIP_RESTORE: true
+      APPVEYOR_CACHE_SKIP_SAVE: true
+
+    - BUILDSYSTEM: cygwin64-gcc
+      CC: gcc
+      CXX: g++
+      APPVEYOR_SAVE_CACHE_ON_ERROR: true
+
+for:
+  -
+    matrix:
+      only:
+        - BUILDSYSTEM: cygwin64-gcc
+
+    install:
+      - ps: >-
+          if ( !(Test-Path "${env:CYG_CACHED_SETUP}" -NewerThan (Get-Date).AddDays(-7)) )
+          {
+            Write-Host "Cached Cygwin setup does not exist or is older than 7 days, downloading from external site."
+            New-Item -Force -Type Directory $env:CYG_CACHE
+            Invoke-WebRequest $env:CYG_URL -OutFile $env:CYG_CACHED_SETUP
+          }
+      - cmd: >-
+          %CYG_CACHED_SETUP% --quiet-mode --no-shortcuts --no-startmenu --no-desktop --upgrade-also --only-site --site "%CYG_MIRROR%" --root "%CYG_ROOT%"  --local-package-dir "%CYG_CACHE%" --packages "%CYG_PACKAGES%"
+
+    # Check that we have a working Cygwin environment before building.
+    before_build:
+      - "%CYG_ROOT%\\bin\\sh -lc 'echo Hello'"
+      - "%CYG_ROOT%\\bin\\sh -lc 'uname -a'"
+
+    build_script:
+      - '%CYG_ROOT%\\bin\\bash -lc "cd $(cygpath ${APPVEYOR_BUILD_FOLDER}) && cd icu4c/source && ./runConfigureICU Cygwin && make check"'
+
+  -
+    matrix:
+      only:
+        - BUILDSYSTEM: VS2017
+
+    build:
+      project: icu4c/source/allinone/allinone.sln
+
+    test_script:
+      - icu4c/source/allinone/icucheck.bat x64 Release