I think there are a few issues here. First, the receipt of the deposit should not affect when you need to claim income or take deductions for the property. This is because the deposit is not "income" to you as you'll eventually have to give it back to your tenant. Second, since the lease started on Jan 1st, it sounds like the property should be placed in service in 2014, which means that you should start putting it on your tax returns for 2014. However, a slight nuance here is that you must have marketed it and shown it in Dec, so depending on the extent of your activity, an argument could be made that it was placed in service in 2013. If so, you will file a schedule for your 2013 tax return with one month of depreciation and no income. This could be a potential benefit to you if you have other passive income. If not, it should be a suspended loss which you can use to offset rental income in the future.
A lot of these issues are fact-specific so I'd recommend talking to a tax advisor. Most of them should be familiar with these relatively common issues. There is also a good IRS publication regarding rental property, which you can find on their website by searching "rental property." None of the comments I've provided in this post should be considered as providing tax or legal advice, but are rather, my own opinion.