Does he not have the power, as President, to overrule the government?
Congress controls the budget. In order to spend money, Donald Trump (or any president) needs Congress to appropriate it.
From the constitution:
The Congress shall have Power To lay and collect Taxes, Duties, Imposts and Excises, to pay the Debts and provide for the common Defence and general Welfare of the United States; but all Duties, Imposts and Excises shall be uniform throughout the United States;
So Congress has the power to budget money (and raise revenue to pay for it, but that's not what we're discussing here). The executive (headed by the president) actually spends the money but only within the confines of the congressional budget.
He could veto a bill to prevent it becoming law. He can interpret existing law in the best light possible (an executive order is an example of this). But he can't actually write law or appropriate money himself. He can only work with what this Congress or some previous Congress has given him.
This is the basic concept of the United States government. There is a separation of powers with checks and balances to preserve it.
In and of itself, a government shutdown doesn't give him much spending authority that he does not already have. Traditionally, it has allowed presidents to determine emergency spending. It's unclear what would happen if he asserted that building the wall was emergency spending. Lots of court cases presumably.
The more normal process is that a shutdown would stop a number of things that Democrats want, e.g. Obamacare payments, other Medicaid payments, welfare payments, etc. So to get those things back, they compromise. And of course, the Republicans have majorities in both chambers of Congress. The question is if a compromise can reach the Senate's supermajority requirement to avoid a filibuster.