<p>I run a game and the running is done by hand, I have a few scripts that help me but essentially it's me doing the work. I am at the moment working on web app that will allow the users to input directly some of their game actions and thus save me a lot of work.</p>

<p>The problem is that I'm one man working on a moderately sized (upwards of 20 tables) project, the workload isn't the issue, it's that bugs will have slipped in even though I test as I write. So my question is thus two-fold.</p>

<ol>
<li>Beta testing, I love open beta's but would a closed beta be somehow more effective and give better results?</li>
<li>How should I bring in the app? Should I one turn drop it in and declare it's being used or should I use it alongside the normal construct of the game?</li>
</ol>

I don't understand what you mean by "bring in the app" and "one turn drop it". By "bring in the app" do you mean deploy? As for "One turn drop", I totally don't understand it.
As for open betas, that depends on your audience, really. Counterstrike, for example, apparently run a few closed betas before doing open betas, so here's my suggestion:
Set up a forum in some free forumboard, or set up a topic in a popular gaming forum.
Look for people (whether or not they are in those forums) that you trust, and let them in in a closed beta. This will allow you to iron out serious kinks at first.
If your closed group isn't reporting as much bugs any more, release it to open beta, pointing out ways on how they could give feedback to you.
This is similar to the approach StackOverflow took, but this being a game setting it up on a gaming forum will give the dual benefit of advertising your game
and
getting some interested beta testers.