The president could simply veto any spending bill until the fiscal year ends. With no new spending authorization, presidents have traditionally just listed some spending as emergency spending. Anything else doesn't get paid until a new authorization is passed, either a short term one or one covering the new fiscal year. Wikipedia
There is no specific dollar amount. Presumably the president could not designate anything as emergency spending and nothing would get spent. Then the dollar amount would be the entire budget for as long as the government was shutdown. Of course, traditionally Congress has issued back pay to the federal employees who were not paid during the crisis, even if they weren't coming to work. Same thing for Social Security and Medicare benefits as well as contracted expenditures.
This approach also requires that the president have the support of at least a third of the members of one chamber of Congress (either the House or the Senate). Otherwise, they could override vetoes. If the president has the support of at least a third of the Senate (thirty-four out of the hundred), then the president would also be safe from removal from office by impeachment.
There is also an argument that the president generally has the power of impoundment, which was used until the Richard Nixon administration. Impoundment means that the president doesn't have to spend all the money that has been authorized in appropriations (spending) bills. The power of impoundment was limited by a law passed during the Nixon administration. There is an argument that such a law is unconstitutional and that the president can generally choose not to spend, although the Supreme Court disagreed in 1975.