The problem of extending Hamiltonian mechanics to include a time
operator, and to interpret a time-energy uncertainty relation, first
posited (without clear formal discussion) in the early days of quantum
mechanics, has a large associated literature; the survey article
P. Busch. The time-energy uncertainty relation, in Time in quantum mechanics (J. Muga et al., eds.), Lecture Notes in Physics vol. 734. Springer, Berlin, 2007. pp 73-105. doi:10.1007/978-3-540-73473-4_3,
carefully reviews the literature up to the year 2000.
(The book in which Busch's survey appears discusses related topics.)
There is no natural operator solution in a Hilbert space setting, as
Pauli showed in 1958,
W. Pauli. Die allgemeinen Prinzipien der Wellenmechanik, in Handbuch der Physik, Vol V/1, p. 60. Springer, Berlin, 1958. Engl. translation: The general principles of quantum mechanics, p. 63. Springer, Berlin 1980.
by a simple argument that a self-adjoint time operator densely defined in a Hilbert space cannot satisfy a CCR with the Hamiltonian, as the CCR would imply that $H$ has as spectrum the whole real line, which is unphysical.
Time measurements do not need a time operator, but are captured well by a positive operator-valued measure (POVM) for the time observable modeling properties of the measuring clock.