I have never seen anything reducing the limits for partial year coverage.
Only income earned while the plan is open can be counted towards the contribution, so the usual advice is to make the plan effective January 1 of the year you open it, which you can do.
The $19k limit is across all plans, so if you already contributed to your previous employer's 401(k) you can't contribute the full $19k to your solo 401(k).
By my math you'd need $195,578 in self employed net income to make the full $56k contribution if you can make the full $19k employee contribution, $291,867 in income if you can't make any of the $19k employee contribution (because you already contributed $19k to another employer's plan), and in between those if you can make some of the employee contribution.
If you open a custom plan (which costs money) that allows for mega backdoor Roth contributions you could hit the limit at a lower income, but I think you'd still need at least as much net income after deducting 1/2 of self employment tax to cover the contribution.
Edit: If you're over 50, there's also a $6k catchup contribution in addition to the $19k/$56k limits.
