Whoa - that looks like an exciting journey to take !
There are direct trains but no direct flights for this journey. kolkata (West Bengal) to Ajmer (Rajasthan) distance is 1642 km and you can get there by flight + cab in just over 7 hours and 42 minutes or by train fares starting from ₹ 536.
The weather at Ajmer is varying from "nice" to "hotter than nice".