Your old printer probably used a USB A/B cable, with the "A" connector at the computer side and the "B" connector at the printer side. Most new printers are designed with home networking in mind and some may no longer support the USB standard. My Brother MFC-L2750DW (all-in-one) supports a USB connector, RJ45, and wireless. Additionally, I have 2 phone jacks for the fax function.
As far as your setup is concerned, is your Lenovo on your home network? If so, is your printer near the router? One option is to connect the printer to your router with an Ethernet cable, and your laptop will then be able to access the printer through the network.
Is wireless not an option for you? That's how mine is setup and I'm quite happy with how it works. With wireless, you are only limited by the reach of your router's signal.
I just took a look and what I am finding is that your new printer does have a USB option. It does not come with a USB cable, but the connector is there. I would take a close look. It's possible that it's covered with a plastic plug.
Just to be clear, the USB cable that your new printer supports will look like this: