Inclusion Criteria:
* Have histologically confirmed adenocarcinoma of the prostate. Primary (or pathologic evidence of conversion to) small cell carcinoma, carcinoid tumor, mixed neuroendocrine carcinoma, large cell neuroendocrine carcinoma, or sarcoma of the prostate is disallowed
* Progressive metastatic androgen pathway modulation-resistant prostate cancer (mAPMR) defined as having a Prostate-Specific Antigen (PSA) level greater than or equal (\>=) 10 nanograms per milliliter (ng/mL) and at least one of the following: (a) PSA progression defined as at least 2 consecutive increases in PSA value over a previous reference value measured at least 1 week apart. (b) Progressive disease or new lesion(s) in the lymph nodes, bones, or viscera on conventional imaging (computed tomography \[CT\]/magnetic resonance imaging \[MRI\]) as defined by response evaluation criteria in solid tumors version 1.1 (RECIST v1.1) and/or on bone scan per Prostate Cancer Working Group 4 (PCWG4) while on Androgen Deprivation Therapy (ADT) (either surgically \[bilateral orchiectomy\] or with medication). Local-regional disease including invasive disease of rectum, bladder, pelvis is allowed as long as participant has a current or past history of distant metastasis (M1 disease). Participants in Part 1 must have measurable disease per RECIST v1.1 by conventional imaging with CT or MRI (chest, abdomen, and pelvis) and/or Technetium-99\^m (99\^mTc) bone scan confirmed by Blinded Independent Central Review (BICR) at the time of screening
* Prior bilateral orchiectomy or medical castration (receiving ongoing ADT with a Gonadotropin-Releasing Hormone (GnRH) analog \[agonist or antagonist\]) with castrate level of serum testosterone (less than \[\<\] 50 nanograms per deciliter \[ng/dL\] or 1.7 nanomoles per liter \[nmol/L\]) prior to the first dose of trial intervention and must continue this therapy throughout the treatment phase
* Have discontinued concurrent use of any non-investigational systemic anticancer therapy (that is, cytotoxic chemotherapy, Androgen Receptor Pathway Inhibitor (ARPI), radiation therapy) within 2 weeks or Lutetium-177 Prostate-Specific Membrane Antigen-617 (177\^Lu PSMA-617) within 6 weeks prior to first dose of trial intervention, or any investigational drugs within 3 months of first dose of trial intervention
* Toxicity related to prior anticancer treatment that is deemed clinically relevant must have resolved to common terminology criteria for adverse events (CTCAE) Version 6.0 Grade 1 or better
Exclusion Criteria:
* Acute or chronic uncontrolled renal disease, pancreatitis, or liver disease (with exception of patients with Gilbert's Syndrome, asymptomatic gallstones, liver metastases, or stable chronic liver disease per investigator assessment)
* Suspected or known allergies, hypersensitivity, or intolerance to JNJ-101556143 or its excipients
* Had major surgery or had significant traumatic injury less than or equal to 28 days of the first dose of trial intervention. Note: Participants with planned surgical procedures to be conducted under local anesthesia may participate
* Spinal cord compression unless considered to have received definitive treatment for this and evidence of clinically stable disease for 28 days
* Solid organ or bone marrow transplantation
