In my opinion, it may be a great idea to call the KMSY Control Tower (I found this number on Google: 504-471-4300) and ask what the busy times are (at least with respect to traffic inbound to KMSY). You might feel more comfortable after speaking to someone at the Tower just to humanize the experience.
I could not locate a phone number for the TRACON, but it's likely you can get a reasonable feel for the busy periods for traffic into the Class B airspace from the KMSY Tower.
Also, it may be a good idea to listen several times a few days prior to your flight to the frequency/radio traffic that you're likely to hear (depending on what direction you are coming from). This way you can familiarize yourself with what you can expect ATC to say to you for clearance into the Class B airspace as well as listen to other pilot's responses. LiveATC.net, which can be used to listen to various frequencies, can be located via this LINK.
Also, the first image below is a listing from LiveATC.net of the MSY TRACON frequencies that may be of value to you.
Below that image is a listing of all of the MSY TRACON frequencies.
I think it's an excellent idea to prepare yourself with as much information as you can. It's likely that your experience into the Class B airspace will be much more enjoyable to you and bolster your confidence for future trips into similar areas.