Wow ! I really wish I could come with you on this one.
There are direct trains but no direct flights for this journey. Travelling by train is the cheapest way of getting there. Bengaluru (Karnataka) to Ajmer (Rajasthan) distance is 2006 km and you can get there by train in just over 43 hours and 35 minutes. However, flight + train tickets come also cheap with fares starting from ₹ 4658.
The heat there will make you sweat, but it's surely bearable.