I don't think that's possible. virsh and virt-manager are both front-ends for controlling libvirtd - the former is a CLI interface, the latter a GUI interface.
You could run something like sudo virsh start "VM-Name" && virt-viewer "VM-Name". And, of course, you could write a shell function or script to do that. For example:
#!/bin/bash
sudo virsh start "$1" && sudo virt-viewer "$1"
Note that this won't start virt-manager itself, it will just start virt-viewer (which is very similar to virt-manager's GUI console for your VM, but lacking several useful features like menu options or buttons to Reboot or Power Off the VM). If you wanted to make use of virt-manager's other features, you'd have to run that separately.
BTW, it's possible to run virsh and virt-viewer without sudo, depending on how libvirtd is configured on your system, and what group(s) your user account is a member of.
For example: on my Debian system, my user account (cas) is a member of the libvirt group. That allows me to run the following:
virsh --connect qemu:///system start VM-NAME &&
virt-viewer --connect qemu:///system VM-NAME
You've mentioned arch-linux in your tags, so you should read https://wiki.archlinux.org/title/Libvirt for details on how libvird is set up on arch.