The CTU said the bank, which could be part-funded by levies on other banks' profits, could de-risk projects that have long-term benefits to New Zealand.
It also saw potential for the bank to establish an investment fund for default KiwiSaver members.
As for the Ministry of Green Works proposal, the CTU said this department could act as an automatic stabiliser in the housing market by building more houses when the construction pipeline lags demand and pulling back when supply from the market is sufficient.
It suggested the ministry could be a "vehicle for central and local government to insource currently outsourced development activities", and separately, be responsible for ensuring Three Waters infrastructure is delivered.
The CTU had a number of ideas on the employment relations front.
It proposed a law change to lift the bar around what constitutes "contracting" to better protect workers who miss out on entitlements due to being classified as a contractor.
The CTU suggested decoupling parental leave so each partner's leave is independent.
"This helps to break down gender segregation in the workplace and reduces the gender pay gap as a result," the CTU said.
"Sweden, Iceland, and Norway have one of the most equitable parental leave schemes which has led to fathers using over 45 per cent, 45 per cent, and 40 per cent of the available parental leave benefits, respectively."
The CTU suggested doing a big trial of a four-day work week.
"This would be undertaken at a sector level, on the same 100/80/100 basis as the UK," it said.
It also recommended progressively extending the provision of public early childhood education services, so it could be free to all children, and making full sick leave and bereavement leave available to employees from their first day of employment.
On the energy front, the CTU wanted to see further steps taken to encourage a switch to cleaner energy.
It added a suggestion to, "Change the way the electricity market works to ensure the amount of electricity suppliers are paid is relative to their generating costs. Instead of the current system of an arbitrarily high price set by one generator usually a fossil fuel producer."
Finally, the CTU said government departments that do cost-benefit analysis of policy proposals should also highlight the cost of not investing. The CTU said they should estimate "the 'liability' the Crown faces for future expenditure set alongside the debt and operating balance figures".
The CTU has already worked closely with the Government and lobby group, BusinessNZ, to a create a Fair Pay Agreements scheme, and is helping design a state income insurance scheme.
It recognised these initiatives in its strategy, but said, "[T]here is still much more to be done…
"[T]he long-term economic problems facing New Zealand are the problems that existed before the pandemic. We face a recovery from Covid-19 that is increasingly K-shaped.
"People with skills, in secure work, and with assets, will continue to do well as the economy comes back from Covid-19. People with insecure work, fewer skills, and low incomes will increasingly feel the impacts of inflation, face housing stress, and a more insecure future."
The CTU said it wanted the Government to tackle the "interrelated problems of productivity, investment, climate change, infrastructure, taxation, jobs, income, equity and wellbeing".
The strategy was launched on Tuesday morning at an event Reserve Bank Governor Adrian Orr spoke at.
