普通はこんな感じで行ける。
http://www.playframework.com/documentation/2.1.3/Migration

project/build.properties
project/plugins.sbt
この２つのバージョン番号書き換えによるバージョンアップ。
あとはclean, runでうまくいく予定だった。

でもsbt 0.13.0, Scala 2.10.3 の組み合わせでの
2.1.3 -> 2.2.0のときは下記のエラーが出て、苦しんだ。

[warn] ==== public: tried
[warn] http://repo1.maven.org/maven2/play/sbt-plugin_2.9.2_0.12/2.2.0/sbt-plugin-2.2.0.pom
[warn] ::::::::::::::::::::::::::::::::::::::::::::::
[warn] :: UNRESOLVED DEPENDENCIES ::
[warn] ::::::::::::::::::::::::::::::::::::::::::::::
[warn] :: play#sbt-plugin;2.2.0: not found
[warn] ::::::::::::::::::::::::::::::::::::::::::::::
[warn]
[warn] Note: Some unresolved dependencies have extra attributes. Check that these dependencies exist with the requested attributes.
[warn] play:sbt-plugin:2.2.0 (sbtVersion=0.12, scalaVersion=2.9.2)
[warn]
sbt.ResolveException: unresolved dependency: play#sbt-plugin;2.2.0: not found

http://stackoverflow.com/questions/13737926/unresolved-dependency-sbt-org-scala-sbtsbt-2-9-10-12-1-not-found-play-exampl
http://stackoverflow.com/questions/19045124/upgrading-project-to-version-2-2-0-of-the-play-framework
この辺りが参考になりますが
結論としては

project/plugins.sbt

addSbtPlugin("com.typesafe.play" % "sbt-plugin" % "2.2.0")

project/build.properties

sbt.version=0.13.0

あとは

play clean
play ~run

これでいけました。