Cron function set to run once per month not completing
I always struggled to get cron jobs running at the right time until I found Elysia Cron. Inside the configuration screen of Elysia Cron you can specify which cron hooks run when, so the standard hooks can be set to run each hour, others each day, each week, each month, etc. and you don't need to worry about coding the date as you've done.

EDIT: I just ran the following code:

$day_of_month_to_run = 01;
$date_string = date('Y-m-'.$day_of_month_to_run);
dpm($date_string);
dpm(date('Y-m-d'));


The results were 2015-04-1 &amp; 2015-04-15 your 01 is being collapsed to 1, you need to change it to '01'