Is this still allowed, whether federally or on the state level? Can you be arrested the day a law is enacted?
The Act was enacted on August 2, 1937. Pub. 238, 75th Congress, 50 Stat. 551
It had an effective date of October 1, 1937, specified at Sec. 17:
This Act shall take effect on the first day of the second month after the month during which it is enacted.
As soon as a law defining a crime is in effect, you can be arrested for that crime.
Calder v. Bull, 3 U.S. 386 (1798) sets out the limits on ex post facto laws, as prohibited by clause 3 of Article I, Section 9 of the Constitution ("No Bill of Attainder or ex post facto Law shall be passed").
A law is prohibited if it "makes an action done before the passing of the law and which was innocent when done, criminal and punishes such action." Thus, the limit is a strict limit against retroactivity, but a law can become effective immediately.
Justice Story, in In re Richardson et al. 2 Story, 571 (1843) wrote:
[...] it appears to me, that in all cases of public laws, the very time of the approval constitutes, and should constitute, the guide as to the time, when the law is to have its effect [...]
The legislature have it in their power to prescribe the very moment, in futuro, after the approval, when a law shall have effect.